Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 14

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Murray R. Spiegel: Mathematical Handbook of Formulas and Tables: Chapter 14

Published $1968$.


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$14 \quad$ Indefinite Integrals

General Rules of Integration

In the following, $u, v, w$ are functions of $x$; $a, b, p, q, n$ any constants, restricted if indicated; $e = 2.71828 \ldots$ is the natural base of logarithms; $\ln u$ denotes the natural logarithm of $u$ where it is assumed that $u > 0$ [in general, to extend formulas to cases where $u < 0$ as well, replace $\ln u$ with $\ln \size u$]; all angles are in radians; all constants of integration are omitted but implied.

$14.1$: Primitive of $a$
$14.2$: Primitive of $a \map f x$
$14.3$: Linear Combination of Integrals: $\displaystyle \int \paren {u \pm v \pm w \pm \cdots} \rd x$
$14.4$: Integration by Parts: $\displaystyle \int u \rd v$
$14.5$: Primitive of $\map f {a x}$
$14.6$: Primitive of $\map F { {\map f x} }$
$14.7$: Primitive of $u^n$
$14.8$: Primitive of $\dfrac {\rd u} u$
$14.9$: Primitive of $e^u$
$14.10$: Primitive of $a^u$
$14.11$: Primitive of $\sin u$
$14.12$: Primitive of $\cos u$
$14.13$: Primitive of $\tan u$
$14.14$: Primitive of $\cot u$
$14.15$: Primitive of $\sec u$
$14.16$: Primitive of $\csc u$
$14.17$: Primitive of $\sec^2 u$
$14.18$: Primitive of $\csc^2 u$
$14.19$: Primitive of $\tan^2 u$
$14.20$: Primitive of $\cot^2 u$
$14.21$: Primitive of $\sin^2 u$
$14.22$: Primitive of $\cos^2 u$
$14.23$: Primitive of $\sec u \tan u$
$14.24$: Primitive of $\csc u \cot u$
$14.25$: Primitive of $\sinh u$
$14.26$: Primitive of $\cosh u$
$14.27$: Primitive of $\tanh u$
$14.28$: Primitive of $\coth u$
$14.29$: Primitive of $\sech u$
$14.30$: Primitive of $\csch u$
$14.31$: Primitive of $\sech^2 u$
$14.32$: Primitive of $\csch^2 u$
$14.33$: Primitive of $\tanh^2 u$
$14.34$: Primitive of $\coth^2 u$
$14.35$: Primitive of $\sinh^2 u$
$14.36$: Primitive of $\cosh^2 u$
$14.37$: Primitive of $\operatorname{sech} u \tanh u$
$14.38$: Primitive of $\operatorname{csch} u \coth u$
$14.39$: Primitive of $\dfrac 1 {u^2 + a^2}$
$14.40$: Primitive of $\dfrac 1 {u^2 - a^2}$
$14.41$: Primitive of $\dfrac 1 {a^2 - u^2}$
$14.42$: Primitive of $\dfrac 1 {\sqrt{a^2 - u^2}}$
$14.43$: Primitive of $\dfrac 1 {\sqrt{u^2 + a^2}}$
$14.44$: Primitive of $\dfrac 1 {\sqrt{u^2 - a^2}}$
$14.45$: Primitive of $\dfrac 1 {u \sqrt{u^2 - a^2}}$
$14.46$: Primitive of $\dfrac 1 {u \sqrt{u^2 + a^2}}$
$14.47$: Primitive of $\dfrac 1 {u \sqrt{a^2 - u^2}}$
$14.48$: Generalized Integration by Parts: $\displaystyle \int f^{\paren n} g \rd x$


Important Transformations

Often in practice an integral can be simplified by using an appropriate transformation or substitution and formula $14.6$. The following list gives some transformations and their effects.

$14.49$: Primitive of $F \left({a x + b}\right)$
$14.50$: Primitive of $F \left({ \sqrt{a x + b} }\right)$
$14.51$: Primitive of $F \left({ \sqrt [n] {a x + b} }\right)$
$14.52$: Primitive of $F \left({ \sqrt{a^2 - x^2} }\right)$
$14.53$: Primitive of $F \left({ \sqrt{x^2 + a^2} }\right)$
$14.54$: Primitive of $F \left({ \sqrt{x^2 - a^2} }\right)$
$14.55$: Primitive of $F \left({e^{a x}}\right)$
$14.56$: Primitive of $F \left({\ln x}\right)$
$14.57$: Primitive of $F \left({\sin^{-1} \dfrac x a}\right)$
Similar results apply for other inverse trigonometric functions:
Primitive of $F \left({\cos^{-1} \dfrac x a}\right)$
Primitive of $F \left({\tan^{-1} \dfrac x a}\right)$
Primitive of $F \left({\cot^{-1} \dfrac x a}\right)$
Primitive of $F \left({\sec^{-1} \dfrac x a}\right)$
Primitive of $F \left({\csc^{-1} \dfrac x a}\right)$
$14.58$: Primitive of $F \left({\sin x, \cos x}\right)$


Special Integrals

It is assumed in all cases that division by zero is excluded. Also, in the following, $u, v, w$ are functions of $x$; $a, b, p, q, n$ any constants, restricted if indicated; $e = 2.71828 \ldots$ is the natural base of logarithms; $\ln u$ denotes the natural logarithm of $u$ where it is assumed that $u > 0$ [in general, to extend formulas to cases where $u < 0$ as well, replace $\ln u$ with $\ln \left\vert{u}\right\vert$]; all angles are in radians; all constants of integration are omitted but implied.

Integrals Involving $a x + b$

$14.59$: Primitive of $\dfrac 1 {a x + b}$
$14.60$: Primitive of $\dfrac x {a x + b}$
$14.61$: Primitive of $\dfrac {x^2} {a x + b}$
$14.62$: Primitive of $\dfrac {x^3} {a x + b}$
$14.63$: Primitive of $\dfrac 1 {x \left({a x + b}\right)}$
$14.64$: Primitive of $\dfrac 1 {x^2 \left({a x + b}\right)}$
$14.65$: Primitive of $\dfrac 1 {x^3 \left({a x + b}\right)}$
$14.66$: Primitive of $\dfrac 1 {\left({a x + b}\right)^2}$
$14.67$: Primitive of $\dfrac x {\left({a x + b}\right)^2}$
$14.68$: Primitive of $\dfrac {x^2} {\left({a x + b}\right)^2}$
$14.69$: Primitive of $\dfrac {x^3} {\left({a x + b}\right)^2}$
$14.70$: Primitive of $\dfrac 1 {x \left({a x + b}\right)^2}$
$14.71$: Primitive of $\dfrac 1 {x^2 \left({a x + b}\right)^2}$
$14.72$: Primitive of $\dfrac 1 {x^3 \left({a x + b}\right)^2}$
$14.73$: Primitive of $\dfrac 1 {\left({a x + b}\right)^3}$
$14.74$: Primitive of $\dfrac x {\left({a x + b}\right)^3}$
$14.75$: Primitive of $\dfrac {x^2} {\left({a x + b}\right)^3}$
$14.76$: Primitive of $\dfrac {x^3} {\left({a x + b}\right)^3}$
$14.77$: Primitive of $\dfrac 1 {x \left({a x + b}\right)^3}$
$14.78$: Primitive of $\dfrac 1 {x^2 \left({a x + b}\right)^3}$
$14.79$: Primitive of $\dfrac 1 {x^3 \left({a x + b}\right)^3}$
$14.80$: Primitive of $\left({a x + b}\right)^n$
$14.81$: Primitive of $x \left({a x + b}\right)^n$
$14.82$: Primitive of $x^2 \left({a x + b}\right)^n$
$14.83$: Primitive of $x^m \left({a x + b}\right)^n$:
Three formulas are given for this case:
Decrement of Power of $a x + b$
Decrement of Power of $x$
Increment of Power of $a x + b$
The fourth form is not given:
Increment of Power of $x$


Integrals Involving $\sqrt{a x + b}$

$14.84$: Primitive of $\dfrac 1 {\sqrt{a x + b}}$
$14.85$: Primitive of $\dfrac x {\sqrt{a x + b}}$
$14.86$: Primitive of $\dfrac {x^2} {\sqrt{a x + b}}$
$14.87$: Primitive of $\dfrac 1 {x \sqrt{a x + b}}$
$14.88$: Primitive of $\dfrac 1 {x^2 \sqrt{a x + b}}$
$14.89$: Primitive of $\sqrt{a x + b}$
$14.90$: Primitive of $x \sqrt{a x + b}$
$14.91$: Primitive of $x^2 \sqrt{a x + b}$
$14.92$: Primitive of $\dfrac {\sqrt{a x + b}} x$
$14.93$: Primitive of $\dfrac {\sqrt{a x + b}} {x^2}$
$14.94$: Primitive of $\dfrac {x^m} {\sqrt{a x + b}}$
$14.95$: Primitive of $\dfrac 1 {x^m \sqrt{a x + b}}$
$14.96$: Primitive of $x^m \sqrt{a x + b}$
$14.97-98$: Primitive of $\dfrac {\sqrt{a x + b}} {x^m}$
Two formulations are given:
$14.97$: Formulation 1
$14.98$: Formulation 2
$14.99$: Primitive of $\left({a x + b}\right)^{m / 2}$
$14.100$: Primitive of $x \left({a x + b}\right)^{m / 2}$
$14.101$: Primitive of $x^2 \left({a x + b}\right)^{m / 2}$
$14.102$: Primitive of $\dfrac {\left({a x + b}\right)^{m / 2}} x$
$14.103$: Primitive of $\dfrac {\left({a x + b}\right)^{m / 2} } {x^2}$
$14.104$: Primitive of $\dfrac 1 {x \left({a x + b}\right)^{m / 2} }$


Integrals Involving $a x + b$ and $p x + q$

$14.105$: Primitive of $\dfrac 1 {\left({a x + b}\right) \left({p x + q}\right)}$
$14.106$: Primitive of $\dfrac x {\left({a x + b}\right) \left({p x + q}\right)}$
$14.107$: Primitive of $\dfrac 1 {\left({a x + b}\right)^2 \left({p x + q}\right)}$
$14.108$: Primitive of $\dfrac x {\left({a x + b}\right)^2 \left({p x + q}\right)}$
$14.109$: Primitive of $\dfrac {x^2} {\left({a x + b}\right)^2 \left({p x + q}\right)}$
$14.110$: Primitive of $\dfrac 1 {\left({a x + b}\right)^m \left({p x + q}\right)^n}$
$14.111$: Primitive of $\dfrac {a x + b} {p x + q}$
$14.112$: Primitive of $\dfrac {\left({a x + b}\right)^m} {\left({p x + q}\right)^n}$


Integrals Involving $\sqrt{a x + b}$ and $p x + q$

$14.113$: Primitive of $\dfrac {p x + q} {\sqrt{a x + b}}$
$14.114$: Primitive of $\dfrac 1 {\left({p x + q}\right) \sqrt{a x + b}}$
$14.115$: Primitive of $\dfrac {\sqrt{a x + b}} {p x + q}$
$14.116$: Primitive of $\left({p x + q}\right)^n \sqrt{a x + b}$
$14.117$: Primitive of $\dfrac 1 {\left({p x + q}\right)^n \sqrt{a x + b}}$
$14.118$: Primitive of $\dfrac {\left({p x + q}\right)^n} {\sqrt{a x + b}}$
$14.119$: Primitive of $\dfrac {\sqrt{a x + b}} {\left({p x + q}\right)^n}$


Integrals Involving $\sqrt{a x + b}$ and $\sqrt{p x + q}$

$14.120$: Primitive of $\dfrac 1 {\sqrt{\left({a x + b}\right) \left({p x + q}\right)}}$
$14.121$: Primitive of $\dfrac x {\sqrt{\left({a x + b}\right) \left({p x + q}\right)}}$
$14.122$: Primitive of $\sqrt{\left({a x + b}\right) \left({p x + q}\right)}$
$14.123$: Primitive of $\sqrt{\dfrac {\left({a x + b}\right)} {\left({p x + q}\right)} }$
$14.124$: Primitive of $\dfrac 1 {\left({p x + q}\right) \sqrt{\left({a x + b}\right) \left({p x + q}\right)}}$


Integrals Involving $x^2 + a^2$

$14.125$: Primitive of $\dfrac 1 {x^2 + a^2}$
$14.126$: Primitive of $\dfrac x {x^2 + a^2}$
$14.127$: Primitive of $\dfrac {x^2} {x^2 + a^2}$
$14.128$: Primitive of $\dfrac {x^3} {x^2 + a^2}$
$14.129$: Primitive of $\dfrac 1 {x \left({x^2 + a^2}\right)}$
$14.130$: Primitive of $\dfrac 1 {x^2 \left({x^2 + a^2}\right)}$
$14.131$: Primitive of $\dfrac 1 {x^3 \left({x^2 + a^2}\right)}$
$14.132$: Primitive of $\dfrac 1 {\left({x^2 + a^2}\right)^2}$
$14.133$: Primitive of $\dfrac x {\left({x^2 + a^2}\right)^2}$
$14.134$: Primitive of $\dfrac {x^2} {\left({x^2 + a^2}\right)^2}$
$14.135$: Primitive of $\dfrac {x^3} {\left({x^2 + a^2}\right)^2}$
$14.136$: Primitive of $\dfrac 1 {x \left({x^2 + a^2}\right)^2}$
$14.137$: Primitive of $\dfrac 1 {x^2 \left({x^2 + a^2}\right)^2}$
$14.138$: Primitive of $\dfrac 1 {x^3 \left({x^2 + a^2}\right)^2}$
$14.140$: Primitive of $\dfrac 1 {\left({x^2 + a^2}\right)^n}$
$14.141$: Primitive of $\dfrac 1 {x \left({x^2 + a^2}\right)^n}$
$14.142$: Primitive of $\dfrac {x^m} {\left({x^2 + a^2}\right)^n}$
$14.143$: Primitive of $\dfrac 1 {x^m \left({x^2 + a^2}\right)^n}$


Integrals Involving $x^2 - a^2$, $x^2 > a^2$

$14.144$: Primitive of $\dfrac 1 {x^2 - a^2}$
$14.145$: Primitive of $\dfrac x {x^2 - a^2}$
$14.146$: Primitive of $\dfrac {x^2} {x^2 - a^2}$
$14.147$: Primitive of $\dfrac {x^3} {x^2 - a^2}$
$14.148$: Primitive of $\dfrac 1 {x \left({x^2 - a^2}\right)}$
$14.149$: Primitive of $\dfrac 1 {x^2 \left({x^2 - a^2}\right)}$
$14.150$: Primitive of $\dfrac 1 {x^3 \left({x^2 - a^2}\right)}$
$14.151$: Primitive of $\dfrac 1 {\left({x^2 - a^2}\right)^2}$
$14.152$: Primitive of $\dfrac x {\left({x^2 - a^2}\right)^2}$
$14.153$: Primitive of $\dfrac {x^2} {\left({x^2 - a^2}\right)^2}$
$14.154$: Primitive of $\dfrac {x^3} {\left({x^2 - a^2}\right)^2}$
$14.155$: Primitive of $\dfrac 1 {x \left({x^2 - a^2}\right)^2}$
$14.156$: Primitive of $\dfrac 1 {x^2 \left({x^2 - a^2}\right)^2}$
$14.157$: Primitive of $\dfrac 1 {x^3 \left({x^2 - a^2}\right)^2}$
$14.158$: Primitive of $\dfrac 1 {\left({x^2 - a^2}\right)^n}$
$14.159$: Primitive of $\dfrac x {\left({x^2 - a^2}\right)^n}$
$14.160$: Primitive of $\dfrac 1 {x \left({x^2 - a^2}\right)^n}$
$14.161$: Primitive of $\dfrac {x^m} {\left({x^2 - a^2}\right)^n}$
$14.162$: Primitive of $\dfrac 1 {x^m \left({x^2 - a^2}\right)^n}$


Integrals Involving $a^2 - x^2$, $x^2 < a^2$

$14.163$: Primitive of $\dfrac 1 {a^2 - x^2}$
$14.164$: Primitive of $\dfrac x {a^2 - x^2}$
$14.165$: Primitive of $\dfrac {x^2} {a^2 - x^2}$
$14.166$: Primitive of $\dfrac {x^3} {a^2 - x^2}$
$14.167$: Primitive of $\dfrac 1 {x \left({a^2 - x^2}\right)}$
$14.168$: Primitive of $\dfrac 1 {x^2 \left({a^2 - x^2}\right)}$
$14.169$: Primitive of $\dfrac 1 {x^3 \left({a^2 - x^2}\right)}$
$14.170$: Primitive of $\dfrac 1 {\left({a^2 - x^2}\right)^2}$
$14.171$: Primitive of $\dfrac x {\left({a^2 - x^2}\right)^2}$
$14.172$: Primitive of $\dfrac {x^2} {\left({a^2 - x^2}\right)^2}$
$14.173$: Primitive of $\dfrac {x^3} {\left({a^2 - x^2}\right)^2}$
$14.174$: Primitive of $\dfrac 1 {x \left({a^2 - x^2}\right)^2}$
$14.175$: Primitive of $\dfrac 1 {x^2 \left({a^2 - x^2}\right)^2}$
$14.176$: Primitive of $\dfrac 1 {x^3 \left({a^2 - x^2}\right)^2}$
$14.177$: Primitive of $\dfrac 1 {\left({a^2 - x^2}\right)^n}$
$14.178$: Primitive of $\dfrac x {\left({a^2 - x^2}\right)^n}$
$14.179$: Primitive of $\dfrac 1 {x \left({a^2 - x^2}\right)^n}$
$14.180$: Primitive of $\dfrac {x^m} {\left({a^2 - x^2}\right)^n}$
$14.181$: Primitive of $\dfrac 1 {x^m \left({a^2 - x^2}\right)^n}$


Integrals Involving $\sqrt{x^2 + a^2}$

$14.182$: Primitive of $\dfrac 1 {\sqrt{x^2 + a^2}}$
$14.183$: Primitive of $\dfrac x {\sqrt{x^2 + a^2}}$
$14.184$: Primitive of $\dfrac {x^2} {\sqrt{x^2 + a^2}}$
$14.185$: Primitive of $\dfrac {x^3} {\sqrt{x^2 + a^2}}$
$14.186$: Primitive of $\dfrac 1 {x \left({\sqrt{x^2 + a^2}}\right)}$
$14.187$: Primitive of $\dfrac 1 {x^2 \left({\sqrt{x^2 + a^2}}\right)}$
$14.188$: Primitive of $\dfrac 1 {x^3 \left({\sqrt{x^2 + a^2}}\right)}$
$14.189$: Primitive of $\sqrt{x^2 + a^2}$
$14.190$: Primitive of $x \sqrt{x^2 + a^2}$
$14.191$: Primitive of $x^2 \sqrt{x^2 + a^2}$
$14.192$: Primitive of $x^3 \sqrt{x^2 + a^2}$
$14.193$: Primitive of $\dfrac {\sqrt{x^2 + a^2}} x$
$14.194$: Primitive of $\dfrac {\sqrt{x^2 + a^2}} {x^2}$
$14.195$: Primitive of $\dfrac {\sqrt{x^2 + a^2}} {x^3}$
$14.196$: Primitive of $\dfrac 1 {\left({x^2 + a^2}\right)^{3/2}}$
$14.197$: Primitive of $\dfrac x {\left({x^2 + a^2}\right)^{3/2}}$
$14.198$: Primitive of $\dfrac {x^2} {\left({x^2 + a^2}\right)^{3/2}}$
$14.199$: Primitive of $\dfrac {x^3} {\left({x^2 + a^2}\right)^{3/2}}$
$14.200$: Primitive of $\dfrac 1 {x \left({x^2 + a^2}\right)^{3/2}}$
$14.201$: Primitive of $\dfrac 1 {x^2 \left({x^2 + a^2}\right)^{3/2}}$
$14.202$: Primitive of $\dfrac 1 {x^3 \left({x^2 + a^2}\right)^{3/2}}$
$14.203$: Primitive of $\left({x^2 + a^2}\right)^{3/2}$
$14.204$: Primitive of $x \left({x^2 + a^2}\right)^{3/2}$
$14.205$: Primitive of $x^2 \left({x^2 + a^2}\right)^{3/2}$
$14.206$: Primitive of $x^3 \left({x^2 + a^2}\right)^{3/2}$
$14.207$: Primitive of $\dfrac {\left({x^2 + a^2}\right)^{3/2}} x$
$14.208$: Primitive of $\dfrac {\left({x^2 + a^2}\right)^{3/2}} {x^2}$
$14.209$: Primitive of $\dfrac {\left({x^2 + a^2}\right)^{3/2}} {x^3}$


Integrals Involving $\sqrt{x^2 - a^2}$

$14.210.1$: Primitive of $\dfrac 1 {\sqrt{x^2 - a^2}}$
$14.210.2$: Primitive of $\dfrac x {\sqrt{x^2 - a^2}}$
$14.211$: Primitive of $\dfrac {x^2} {\sqrt{x^2 - a^2}}$
$14.212$: Primitive of $\dfrac {x^3} {\sqrt{x^2 - a^2}}$
$14.213$: Primitive of $\dfrac 1 {x \left({\sqrt{x^2 - a^2}}\right)}$
$14.214$: Primitive of $\dfrac 1 {x^2 \left({\sqrt{x^2 - a^2}}\right)}$
$14.215$: Primitive of $\dfrac 1 {x^3 \left({\sqrt{x^2 - a^2}}\right)}$
$14.216$: Primitive of $\sqrt{x^2 - a^2}$
$14.217$: Primitive of $x \sqrt{x^2 - a^2}$
$14.218$: Primitive of $x^2 \sqrt{x^2 - a^2}$
$14.219$: Primitive of $x^3 \sqrt{x^2 - a^2}$
$14.220$: Primitive of $\dfrac {\sqrt{x^2 - a^2}} x$
$14.221$: Primitive of $\dfrac {\sqrt{x^2 - a^2}} {x^2}$
$14.222$: Primitive of $\dfrac {\sqrt{x^2 - a^2}} {x^3}$
$14.223$: Primitive of $\dfrac 1 {\left({x^2 - a^2}\right)^{3/2}}$
$14.224$: Primitive of $\dfrac x {\left({x^2 - a^2}\right)^{3/2}}$
$14.225$: Primitive of $\dfrac {x^2} {\left({x^2 - a^2}\right)^{3/2}}$
$14.226$: Primitive of $\dfrac {x^3} {\left({x^2 - a^2}\right)^{3/2}}$
$14.227$: Primitive of $\dfrac 1 {x \left({x^2 - a^2}\right)^{3/2}}$
$14.228$: Primitive of $\dfrac 1 {x^2 \left({x^2 - a^2}\right)^{3/2}}$
$14.229$: Primitive of $\dfrac 1 {x^3 \left({x^2 - a^2}\right)^{3/2}}$
$14.230$: Primitive of $\left({x^2 - a^2}\right)^{3/2}$
$14.231$: Primitive of $x \left({x^2 - a^2}\right)^{3/2}$
$14.232$: Primitive of $x^2 \left({x^2 - a^2}\right)^{3/2}$
$14.233$: Primitive of $x^3 \left({x^2 - a^2}\right)^{3/2}$
$14.234$: Primitive of $\dfrac {\left({x^2 - a^2}\right)^{3/2}} x$
$14.235$: Primitive of $\dfrac {\left({x^2 - a^2}\right)^{3/2}} {x^2}$
$14.236$: Primitive of $\dfrac {\left({x^2 - a^2}\right)^{3/2}} {x^3}$


Integrals Involving $\sqrt{a^2 - x^2}$

$14.237$: Primitive of $\dfrac 1 {\sqrt{a^2 - x^2}}$
$14.238$: Primitive of $\dfrac x {\sqrt{a^2 - x^2}}$
$14.239$: Primitive of $\dfrac {x^2} {\sqrt{a^2 - x^2}}$
$14.240$: Primitive of $\dfrac {x^3} {\sqrt{a^2 - x^2}}$
$14.241$: Primitive of $\dfrac 1 {x \left({\sqrt{a^2 - x^2}}\right)}$
$14.242$: Primitive of $\dfrac 1 {x^2 \left({\sqrt{a^2 - x^2}}\right)}$
$14.243$: Primitive of $\dfrac 1 {x^3 \left({\sqrt{a^2 - x^2}}\right)}$
$14.244$: Primitive of $\sqrt{a^2 - x^2}$
$14.245$: Primitive of $x \sqrt{a^2 - x^2}$
$14.246$: Primitive of $x^2 \sqrt{a^2 - x^2}$
$14.247$: Primitive of $x^3 \sqrt{a^2 - x^2}$
$14.248$: Primitive of $\dfrac {\sqrt{a^2 - x^2}} x$
$14.249$: Primitive of $\dfrac {\sqrt{a^2 - x^2}} {x^2}$
$14.250$: Primitive of $\dfrac {\sqrt{a^2 - x^2}} {x^3}$
$14.251$: Primitive of $\dfrac 1 {\left({a^2 - x^2}\right)^{3/2}}$
$14.252$: Primitive of $\dfrac x {\left({a^2 - x^2}\right)^{3/2}}$
$14.253$: Primitive of $\dfrac {x^2} {\left({a^2 - x^2}\right)^{3/2}}$
$14.254$: Primitive of $\dfrac {x^3} {\left({a^2 - x^2}\right)^{3/2}}$
$14.255$: Primitive of $\dfrac 1 {x \left({a^2 - x^2}\right)^{3/2}}$
$14.256$: Primitive of $\dfrac 1 {x^2 \left({a^2 - x^2}\right)^{3/2}}$
$14.257$: Primitive of $\dfrac 1 {x^3 \left({a^2 - x^2}\right)^{3/2}}$
$14.258$: Primitive of $\left({a^2 - x^2}\right)^{3/2}$
$14.259$: Primitive of $x \left({a^2 - x^2}\right)^{3/2}$
$14.260$: Primitive of $x^2 \left({a^2 - x^2}\right)^{3/2}$
$14.261$: Primitive of $x^3 \left({a^2 - x^2}\right)^{3/2}$
$14.262$: Primitive of $\dfrac {\left({a^2 - x^2}\right)^{3/2}} x$
$14.263$: Primitive of $\dfrac {\left({a^2 - x^2}\right)^{3/2}} {x^2}$
$14.264$: Primitive of $\dfrac {\left({a^2 - x^2}\right)^{3/2}} {x^3}$


Integrals Involving $a x^2 + b x + c$

$14.265$: Primitive of $\dfrac 1 {a x^2 + b x + c}$


If $b^2 = 4 a c$, $a x^2 + b x + c = a \left({x + b / 2 a}\right)^2$ and the results from Integrals Involving $a x + b$ can be used. If $b = 0$ use results from Integrals Involving $x^2 + a^2$. If $a$ or $c = 0$ use results from Integrals Involving $a x + b$.


$14.266$: Primitive of $\dfrac x {a x^2 + b x + c}$
$14.267$: Primitive of $\dfrac {x^2} {a x^2 + b x + c}$
$14.268$: Primitive of $\dfrac {x^m} {a x^2 + b x + c}$
$14.269$: Primitive of $\dfrac 1 {x \left({a x^2 + b x + c}\right)}$
$14.270$: Primitive of $\dfrac 1 {x^2 \left({a x^2 + b x + c}\right)}$
$14.271$: Primitive of $\dfrac 1 {x^n \left({a x^2 + b x + c}\right)}$
$14.272$: Primitive of $\dfrac 1 {\left({a x^2 + b x + c}\right)^2}$
$14.273$: Primitive of $\dfrac x {\left({a x^2 + b x + c}\right)^2}$
$14.274$: Primitive of $\dfrac {x^2} {\left({a x^2 + b x + c}\right)^2}$
$14.275$: Primitive of $\dfrac {x^m} {\left({a x^2 + b x + c}\right)^n}$
$14.276$: Primitive of $\dfrac {x^{2 n - 1}} {\left({a x^2 + b x + c}\right)^n}$
$14.277$: Primitive of $\dfrac 1 {x \left({a x^2 + b x + c}\right)^2}$
$14.278$: Primitive of $\dfrac 1 {x^2 \left({a x^2 + b x + c}\right)^2}$
$14.279$: Primitive of $\dfrac 1 {x^m \left({a x^2 + b x + c}\right)^n}$


Integrals Involving $\sqrt {a x^2 + b x + c}$

In the following results if $b^2 = 4 a c$, $\sqrt{a x^2 + b x + c} = \sqrt a \left({x + b / 2 a}\right)$ and the results from Integrals Involving $a x + b$ can be used. If $b = 0$ use results from Integrals Involving $\sqrt {x^2 + a^2}$, Integrals Involving $\sqrt {x^2 - a^2}$, $x^2 > a^2$ and Integrals Involving $\sqrt {a^2 - x^2}$, $a^2 > x^2$. If $a$ or $c = 0$ use results from Integrals Involving $\sqrt{a x + b}$.

$14.280$: Primitive of $\dfrac 1 {\sqrt{a x^2 + b x + c}}$
$14.281$: Primitive of $\dfrac x {\sqrt{a x^2 + b x + c}}$
$14.282$: Primitive of $\dfrac {x^2} {\sqrt{a x^2 + b x + c}}$
$14.283$: Primitive of $\dfrac 1 {x \left({\sqrt{a x^2 + b x + c}}\right)}$
$14.284$: Primitive of $\dfrac 1 {x^2 \left({\sqrt{a x^2 + b x + c}}\right)}$
$14.285$: Primitive of $\sqrt{a x^2 + b x + c}$
$14.286$: Primitive of $x \sqrt{a x^2 + b x + c}$
$14.287$: Primitive of $x^2 \sqrt{a x^2 + b x + c}$
$14.288$: Primitive of $\dfrac {\sqrt{a x^2 + b x + c}} x$
$14.289$: Primitive of $\dfrac {\sqrt{a x^2 + b x + c}} {x^2}$
$14.290$: Primitive of $\dfrac 1 {\left({a x^2 + b x + c}\right)^{3/2}}$
$14.291$: Primitive of $\dfrac x {\left({a x^2 + b x + c}\right)^{3/2}}$
$14.292$: Primitive of $\dfrac {x^2} {\left({a x^2 + b x + c}\right)^{3/2}}$
$14.293$: Primitive of $\dfrac 1 {x \left({a x^2 + b x + c}\right)^{3/2}}$
$14.294$: Primitive of $\dfrac 1 {x^2 \left({a x^2 + b x + c}\right)^{3/2}}$
$14.295$: Primitive of $\left({a x^2 + b x + c}\right)^{n + 1 / 2}$
$14.296$: Primitive of $x \left({a x^2 + b x + c}\right)^{n + 1 / 2}$
$14.297$: Primitive of $\dfrac 1 {\left({a x^2 + b x + c}\right)^{n + 1 / 2}}$
$14.298$: Primitive of $\dfrac 1 {x \left({a x^2 + b x + c}\right)^{n + 1 / 2}}$


Integrals Involving $x^3 + a^3$

Note that for formulas involving $x^3 - a^3$ replace $a$ by $-a$.

$14.299$: Primitive of $\dfrac 1 {x^3 + a^3}$
$14.300$: Primitive of $\dfrac x {x^3 + a^3}$
$14.301$: Primitive of $\dfrac {x^2} {x^3 + a^3}$
$14.302$: Primitive of $\dfrac 1 {x \left({x^3 + a^3}\right)}$
$14.303$: Primitive of $\dfrac 1 {x^2 \left({x^3 + a^3}\right)}$
$14.304$: Primitive of $\dfrac 1 {\left({x^3 + a^3}\right)^2}$
$14.305$: Primitive of $\dfrac x {\left({x^3 + a^3}\right)^2}$
$14.306$: Primitive of $\dfrac {x^2} {\left({x^3 + a^3}\right)^2}$
$14.307$: Primitive of $\dfrac 1 {x \left({x^3 + a^3}\right)^2}$
$14.308$: Primitive of $\dfrac 1 {x^2 \left({x^3 + a^3}\right)^2}$
$14.309$: Primitive of $\dfrac {x^m} {x^3 + a^3}$
$14.310$: Primitive of $\dfrac 1 {x^n \left({x^3 + a^3}\right)}$


Integrals Involving $x^4 \pm a^4$

$14.311$: Primitive of $\dfrac 1 {x^4 + a^4}$
$14.312$: Primitive of $\dfrac x {x^4 + a^4}$
$14.313$: Primitive of $\dfrac {x^2} {x^4 + a^4}$
$14.314$: Primitive of $\dfrac {x^3} {x^4 + a^4}$
$14.315$: Primitive of $\dfrac 1 {x \left({x^4 + a^4}\right)}$
$14.316$: Primitive of $\dfrac 1 {x^2 \left({x^4 + a^4}\right)}$
$14.317$: Primitive of $\dfrac 1 {x^3 \left({x^4 + a^4}\right)}$
$14.318$: Primitive of $\dfrac 1 {x^4 - a^4}$
$14.319$: Primitive of $\dfrac x {x^4 - a^4}$
$14.320$: Primitive of $\dfrac {x^2} {x^4 - a^4}$
$14.321$: Primitive of $\dfrac {x^3} {x^4 - a^4}$
$14.322$: Primitive of $\dfrac 1 {x \left({x^4 - a^4}\right)}$
$14.323$: Primitive of $\dfrac 1 {x^2 \left({x^4 - a^4}\right)}$
$14.324$: Primitive of $\dfrac 1 {x^3 \left({x^4 - a^4}\right)}$


Integrals Involving $x^n \pm a^n$

$14.325$: Primitive of $\dfrac 1 {x \left({x^n + a^n}\right)}$
$14.326$: Primitive of $\dfrac {x^{n - 1}} {x \left({x^n + a^n}\right)}$
$14.327$: Primitive of $\dfrac {x^m} {\left({x^n + a^n}\right)^r}$
$14.328$: Primitive of $\dfrac 1 {x^m \left({x^n + a^n}\right)^r}$
$14.329$: Primitive of $\dfrac 1 {x \sqrt{x^n + a^n}}$
$14.330$: Primitive of $\dfrac 1 {x \left({x^n - a^n}\right)}$
$14.331$: Primitive of $\dfrac {x^{n - 1}} {x \left({x^n - a^n}\right)}$
$14.332$: Primitive of $\dfrac {x^m} {\left({x^n - a^n}\right)^r}$
$14.333$: Primitive of $\dfrac 1 {x^m \left({x^n - a^n}\right)^r}$
$14.334$: Primitive of $\dfrac 1 {x \sqrt{x^n - a^n}}$
$14.335$: Primitive of $\dfrac {x^{p - 1}} {x^{2 m} + a^{2 m}}$
$14.336$: Primitive of $\dfrac {x^{p - 1}} {x^{2 m} - a^{2 m}}$
$14.337$: Primitive of $\dfrac {x^{p - 1}} {x^{2 m + 1} + a^{2 m + 1}}$
$14.338$: Primitive of $\dfrac {x^{p - 1}} {x^{2 m + 1} - a^{2 m + 1}}$


Integrals Involving $\sin a x$

$14.339$: Primitive of $\sin a x$
$14.340$: Primitive of $x \sin a x$
$14.341$: Primitive of $x^2 \sin a x$
$14.342$: Primitive of $x^3 \sin a x$
$14.343$: Primitive of $\dfrac {\sin a x} x$
$14.344$: Primitive of $\dfrac {\sin a x} {x^2}$
$14.345$: Primitive of $\dfrac 1 {\sin a x}$
$14.346$: Primitive of $\dfrac x {\sin a x}$
$14.347$: Primitive of $\sin^2 a x$
$14.348$: Primitive of $x \sin^2 a x$
$14.349$: Primitive of $\sin^3 a x$
$14.350$: Primitive of $\sin^4 a x$
$14.351$: Primitive of $\dfrac 1 {\sin^2 a x}$
$14.352$: Primitive of $\dfrac 1 {\sin^3 a x}$
$14.353$: Primitive of $\sin p x \sin q x$
[If $p = \pm q$, see $14.347$: Primitive of $\sin^2 a x$.]
$14.354$: Primitive of $\dfrac 1 {1 - \sin a x}$
$14.355$: Primitive of $\dfrac x {1 - \sin a x}$
$14.356$: Primitive of $\dfrac 1 {1 + \sin a x}$
$14.357$: Primitive of $\dfrac x {1 + \sin a x}$
$14.358$: Primitive of $\dfrac 1 {\left({1 - \sin a x}\right)^2}$
$14.359$: Primitive of $\dfrac 1 {\left({1 + \sin a x}\right)^2}$
$14.360$: Primitive of $\dfrac 1 {p + q \sin a x}$
[If $p = \pm q$, see $14.354$: Primitive of $\dfrac 1 {1 - \sin a x}$ and $14.356$: Primitive of $\dfrac 1 {1 + \sin a x}$.]
$14.361$: Primitive of $\dfrac 1 {\left({p + q \sin a x}\right)^2}$
[If $p = \pm q$, see $14.358$: Primitive of $\dfrac 1 {\left({1 - \sin a x}\right)^2}$ and $14.359$: Primitive of $\dfrac 1 {\left({1 + \sin a x}\right)^2}$.]
$14.362$: Primitive of $\dfrac 1 {p^2 + q^2 \sin^2 a x}$
$14.363$: Primitive of $\dfrac 1 {p^2 - q^2 \sin^2 a x}$
$14.364$: Primitive of $x^m \sin a x$
$14.365$: Primitive of $\dfrac {\sin a x} {x^n}$
$14.366$: Primitive of $\sin^n a x$
$14.367$: Primitive of $\dfrac 1 {\sin^n a x}$
$14.368$: Primitive of $\dfrac x {\sin^n a x}$


Integrals Involving $\cos a x$

$14.369$: Primitive of $\cos a x$
$14.370$: Primitive of $x \cos a x$
$14.371$: Primitive of $x^2 \cos a x$
$14.372$: Primitive of $x^3 \cos a x$
$14.373$: Primitive of $\dfrac {\cos a x} x$
$14.374$: Primitive of $\dfrac {\cos a x} {x^2}$
$14.375$: Primitive of $\dfrac 1 {\cos a x}$
$14.376$: Primitive of $\dfrac x {\cos a x}$
$14.377$: Primitive of $\cos^2 a x$
$14.378$: Primitive of $x \cos^2 a x$
$14.379$: Primitive of $\cos^3 a x$
$14.380$: Primitive of $\cos^4 a x$
$14.381$: Primitive of $\dfrac 1 {\cos^2 a x}$
$14.382$: Primitive of $\dfrac 1 {\cos^3 a x}$
$14.383$: Primitive of $\cos a x \cos p x$
[If $p = \pm q$, see $14.377$: Primitive of $\cos^2 a x$.]
$14.384$: Primitive of $\dfrac 1 {1 - \cos a x}$
$14.385$: Primitive of $\dfrac x {1 - \cos a x}$
$14.386$: Primitive of $\dfrac 1 {1 + \cos a x}$
$14.387$: Primitive of $\dfrac x {1 + \cos a x}$
$14.388$: Primitive of $\dfrac 1 {\left({1 - \cos a x}\right)^2}$
$14.389$: Primitive of $\dfrac 1 {\left({1 + \cos a x}\right)^2}$
$14.390$: Primitive of $\dfrac 1 {p + q \cos a x}$
[If $p = \pm q$, see $14.384$: Primitive of $\dfrac 1 {1 - \cos a x}$ and $14.386$: Primitive of $\dfrac 1 {1 + \cos a x}$.]
$14.391$: Primitive of $\dfrac 1 {\left({p + q \cos a x}\right)^2}$
[If $p = \pm q$, see $14.388$: Primitive of $\dfrac 1 {\left({1 - \cos a x}\right)^2}$ and $14.389$: Primitive of $\dfrac 1 {\left({1 + \cos a x}\right)^2}$.]
$14.392$: Primitive of $\dfrac 1 {p^2 + q^2 \cos^2 a x}$
$14.393$: Primitive of $\dfrac 1 {p^2 - q^2 \cos^2 a x}$
$14.394$: Primitive of $x^m \cos a x$
$14.395$: Primitive of $\dfrac {\cos a x} {x^n}$
$14.396$: Primitive of $\cos^n a x$
$14.397$: Primitive of $\dfrac 1 {\cos^n a x}$
$14.398$: Primitive of $\dfrac x {\cos^n a x}$


Integrals Involving $\sin a x$ and $\cos a x$

$14.399$: Primitive of $\sin a x \cos a x$
$14.400$: Primitive of $\sin p x \cos q x$
$14.401$: Primitive of $\sin^n a x \cos a x$
$14.402$: Primitive of $\cos^n a x \sin a x$
$14.403$: Primitive of $\sin^2 a x \cos^2 a x$
$14.404$: Primitive of $\dfrac 1 {\sin a x \cos a x}$
$14.405$: Primitive of $\dfrac 1 {\sin^2 a x \cos a x}$
$14.406$: Primitive of $\dfrac 1 {\sin a x \cos^2 a x}$
$14.407$: Primitive of $\dfrac 1 {\sin^2 a x \cos^2 a x}$
$14.408$: Primitive of $\dfrac {\sin^2 a x} {\cos a x}$
$14.409$: Primitive of $\dfrac {\cos^2 a x} {\sin a x}$
$14.410$: Primitive of $\dfrac 1 {\cos a x \left({1 \pm \sin a x}\right)}$
$14.410.1$: Primitive of $\dfrac 1 {\cos a x \left({1 + \sin a x}\right)}$
$14.410.2$: Primitive of $\dfrac 1 {\cos a x \left({1 - \sin a x}\right)}$
$14.411$: Primitive of $\dfrac 1 {\sin a x \left({1 \pm \cos a x}\right)}$
$14.411.1$: Primitive of $\dfrac 1 {\sin a x \left({1 + \cos a x}\right)}$
$14.411.2$: Primitive of $\dfrac 1 {\sin a x \left({1 - \cos a x}\right)}$
$14.412$: Primitive of $\dfrac 1 {\sin a x \pm \cos a x}$
$14.412.1$: Primitive of $\dfrac 1 {\sin a x + \cos a x}$
$14.412.2$: Primitive of $\dfrac 1 {\sin a x \left({1 - \cos a x}\right)}$
$14.413$: Primitive of $\dfrac {\sin a x} {\sin a x \pm \cos a x}$
$14.413.1$: Primitive of $\dfrac {\sin a x} {\sin a x + \cos a x}$
$14.413.2$: Primitive of $\dfrac {\sin a x} {\sin a x \left({1 - \cos a x}\right)}$
$14.414$: Primitive of $\dfrac {\cos a x} {\sin a x \pm \cos a x}$
$14.414.1$: Primitive of $\dfrac {\cos a x} {\sin a x + \cos a x}$
$14.414.2$: Primitive of $\dfrac {\cos a x} {\sin a x \left({1 - \cos a x}\right)}$
$14.415$: Primitive of $\dfrac {\sin a x} {p + q \cos a x}$
$14.416$: Primitive of $\dfrac {\cos a x} {p + q \sin a x}$
$14.417$: Primitive of $\dfrac {\sin a x} {\left({p + q \cos a x}\right)^n}$
$14.418$: Primitive of $\dfrac {\cos a x} {\left({p + q \cos a x}\right)^n}$
$14.419$: Primitive of $\dfrac 1 {p \sin a x + q \cos a x}$
$14.420$: Primitive of $\dfrac 1 {p \sin a x + q \cos a x + r}$
[If $r = q$, see $14.421$: Primitive of $\dfrac 1 {p \sin a x + q \left({1 + \cos a x}\right)}$.
If $r^2 = p^2 + q^2$, see $14.422$: Primitive of $\dfrac 1 {p \sin a x + q \cos a x \pm \sqrt{p^2 + q^2}}$.]
$14.421$: Primitive of $\dfrac 1 {p \sin a x + q \left({1 + \cos a x}\right)}$
$14.422$: Primitive of $\dfrac 1 {p \sin a x + q \cos a x \pm \sqrt{p^2 + q^2}}$
$14.422.1$: Primitive of $\dfrac 1 {p \sin a x + q \cos a x + \sqrt{p^2 + q^2}}$
$14.422.2$: Primitive of $\dfrac 1 {p \sin a x + q \cos a x - \sqrt{p^2 + q^2}}$
$14.423$: Primitive of $\dfrac 1 {p^2 \sin^2 a x + q^2 \cos^2 a x}$
$14.424$: Primitive of $\dfrac 1 {p^2 \sin^2 a x - q^2 \cos^2 a x}$
$14.425$: Primitive of $\sin^m a x \cos^n a x$
$14.425.1$: Reduction of Power of Sine
$14.425.2$: Reduction of Power of Cosine
$14.426$: Primitive of $\dfrac {\sin^m a x} {\cos^n a x}$
$14.426.1$: Reduction of Both Powers
$14.426.2$: Reduction of Power of Sine
$14.426.3$: Reduction of Power of Cosine
$14.427$: Primitive of $\dfrac {\cos^m a x} {\sin^n a x}$
$14.427.1$: Reduction of Both Powers
$14.427.2$: Reduction of Power of Sine
$14.427.3$: Reduction of Power of Cosine
$14.428$: Primitive of $\dfrac 1 {\sin^m a x \cos^n a x}$
$14.428.1$: Reduction of Power of Cosine
$14.428.2$: Reduction of Power of Sine


Integrals Involving $\tan a x$

$14.429$: Primitive of $\tan a x$
$14.430$: Primitive of $\tan^2 a x$
$14.431$: Primitive of $\tan^3 a x$
$14.432$: Primitive of $\tan^n a x \sec^2 a x$
$14.433$: Primitive of $\dfrac {\sec^2 a x} {\tan a x}$
$14.434$: Primitive of $\dfrac 1 {\tan a x}$
$14.435$: Primitive of $x \tan a x$
$14.436$: Primitive of $\dfrac {\tan a x} x$
$14.437$: Primitive of $x \tan^2 a x$
$14.438$: Primitive of $\dfrac 1 {p + q \tan a x}$
$14.439$: Primitive of $\tan^n a x$


Integrals Involving $\cot a x$

$14.440$: Primitive of $\cot a x$
$14.441$: Primitive of $\cot^2 a x$
$14.442$: Primitive of $\cot^3 a x$
$14.443$: Primitive of $\cot^n a x \csc^2 a x$
$14.444$: Primitive of $\dfrac {\csc^2 a x} {\cot a x}$
$14.445$: Primitive of $\dfrac 1 {\cot a x}$
$14.446$: Primitive of $x \cot a x$
$14.447$: Primitive of $\dfrac {\cot a x} x$
$14.448$: Primitive of $x \cot^2 a x$
$14.449$: Primitive of $\dfrac 1 {p + q \cot a x}$
$14.450$: Primitive of $\cot^n a x$


Integrals Involving $\sec a x$

$14.451$: Primitive of $\sec a x$
$14.452$: Primitive of $\sec^2 a x$
$14.453$: Primitive of $\sec^3 a x$
$14.454$: Primitive of $\sec^n a x \tan a x$
$14.455$: Primitive of $\dfrac 1 {\sec a x}$
$14.456$: Primitive of $x \sec a x$
$14.457$: Primitive of $\dfrac {\sec a x} x$
$14.458$: Primitive of $x \sec^2 a x$
$14.459$: Primitive of $\dfrac 1 {q + p \sec a x}$
$14.460$: Primitive of $\sec^n a x$


Integrals Involving $\csc a x$

$14.461$: Primitive of $\csc a x$
$14.462$: Primitive of $\csc^2 a x$
$14.463$: Primitive of $\csc^3 a x$
$14.464$: Primitive of $\csc^n a x \cot a x$
$14.465$: Primitive of $\dfrac 1 {\csc a x}$
$14.466$: Primitive of $x \csc a x$
$14.467$: Primitive of $\dfrac {\csc a x} x$
$14.468$: Primitive of $x \sec^2 a x$
$14.469$: Primitive of $\dfrac 1 {q + p \csc a x}$
$14.470$: Primitive of $\csc^n a x$


Integrals Involving Inverse Trigonometric Functions

$14.471$: Primitive of $\sin^{-1} \dfrac x a$
$14.472$: Primitive of $x \sin^{-1} \dfrac x a$
$14.473$: Primitive of $x^2 \sin^{-1} \dfrac x a$
$14.474$: Primitive of $\dfrac {\sin^{-1} \dfrac x a} x$
$14.475$: Primitive of $\dfrac {\sin^{-1} \dfrac x a} {x^2}$
$14.476$: Primitive of $\left({\sin^{-1} \dfrac x a}\right)^2$
$14.477$: Primitive of $\cos^{-1} \dfrac x a$
$14.478$: Primitive of $x \cos^{-1} \dfrac x a$
$14.479$: Primitive of $x^2 \cos^{-1} \dfrac x a$
$14.480$: Primitive of $\dfrac {\cos^{-1} \dfrac x a} x$
$14.481$: Primitive of $\dfrac {\cos^{-1} \dfrac x a} {x^2}$
$14.482$: Primitive of $\left({\cos^{-1} \dfrac x a}\right)^2$
$14.483$: Primitive of $\tan^{-1} \dfrac x a$
$14.484$: Primitive of $x \tan^{-1} \dfrac x a$
$14.485$: Primitive of $x^2 \tan^{-1} \dfrac x a$
$14.486$: Primitive of $\dfrac {\tan^{-1} \dfrac x a} x$
$14.487$: Primitive of $\dfrac {\tan^{-1} \dfrac x a} {x^2}$
$14.488$: Primitive of $\cot^{-1} \dfrac x a$
$14.489$: Primitive of $x \cot^{-1} \dfrac x a$
$14.490$: Primitive of $x^2 \cot^{-1} \dfrac x a$
$14.491$: Primitive of $\dfrac {\cot^{-1} \dfrac x a} x$
$14.492$: Primitive of $\dfrac {\cot^{-1} \dfrac x a} {x^2}$
$14.493$: Primitive of $\sec^{-1} \dfrac x a$
$14.494$: Primitive of $x \sec^{-1} \dfrac x a$
$14.495$: Primitive of $x^2 \sec^{-1} \dfrac x a$
$14.496$: Primitive of $\dfrac {\sec^{-1} \dfrac x a} x$
$14.497$: Primitive of $\dfrac {\sec^{-1} \dfrac x a} {x^2}$
$14.498$: Primitive of $\csc^{-1} \dfrac x a$
$14.499$: Primitive of $x \csc^{-1} \dfrac x a$
$14.500$: Primitive of $x^2 \csc^{-1} \dfrac x a$
$14.501$: Primitive of $\dfrac {\csc^{-1} \dfrac x a} x$
$14.502$: Primitive of $\dfrac {\csc^{-1} \dfrac x a} {x^2}$
$14.503$: Primitive of $x^m \sin^{-1} \dfrac x a$
$14.504$: Primitive of $x^m \cos^{-1} \dfrac x a$
$14.505$: Primitive of $x^m \tan^{-1} \dfrac x a$
$14.506$: Primitive of $x^m \cot^{-1} \dfrac x a$
$14.507$: Primitive of $x^m \sec^{-1} \dfrac x a$
$14.508$: Primitive of $x^m \csc^{-1} \dfrac x a$


Integrals Involving $e^{a x}$

$14.509$: Primitive of $e^{a x}$
$14.510$: Primitive of $x e^{a x}$
$14.511$: Primitive of $x^2 e^{a x}$
$14.512$: Primitive of $x^n e^{a x}$
$14.513$: Primitive of $\dfrac {e^{a x}} x$
$14.514$: Primitive of $\dfrac {e^{a x}} {x^n}$
$14.515$: Primitive of $\dfrac 1 {p + q e^{a x}}$
$14.516$: Primitive of $\dfrac 1 {\left({p + q e^{a x}}\right)^2}$
$14.517$: Primitive of $\dfrac 1 {p e^{a x} + q e^{-a x}}$
$14.518$: Primitive of $e^{a x} \sin b x$
$14.519$: Primitive of $e^{a x} \cos b x$
$14.520$: Primitive of $x e^{a x} \sin b x$
$14.521$: Primitive of $x e^{a x} \cos b x$
$14.522$: Primitive of $e^{a x} \ln x$
$14.523$: Primitive of $e^{a x} \sin^n b x$
$14.524$: Primitive of $e^{a x} \cos^n b x$


Integrals Involving $\ln x$

$14.525$: Primitive of $\ln x$
$14.526$: Primitive of $x \ln x$
$14.527$: Primitive of $x^m \ln x$
$14.528$: Primitive of $\dfrac {\ln x} x$
$14.529$: Primitive of $\dfrac {\ln x} {x^2}$
$14.530$: Primitive of $\dfrac {\ln x} x$
$14.531$: Primitive of $\dfrac {\ln^n x} x$
[If $n = -1$, see $14.532$: Primitive of $\dfrac 1 {x \ln x}$.]
$14.532$: Primitive of $\dfrac 1 {x \ln x}$
$14.533$: Primitive of $\dfrac 1 {\ln x}$
$14.534$: Primitive of $\dfrac {x^m} {\ln x}$
$14.535$: Primitive of $\ln^n x$
$14.536$: Primitive of $x^m \ln^n x$
$14.537$: Primitive of $\ln \left({x^2 + a^2}\right)$
$14.537$: Primitive of $\ln \left({x^2 - a^2}\right)$
$14.539$: Primitive of $x^m \ln \left({x^2 \pm a^2}\right)$
$14.539.1$: Primitive of $x^m \ln \left({x^2 + a^2}\right)$
$14.539.2$: Primitive of $x^m \ln \left({x^2 - a^2}\right)$


Integrals Involving $\sinh a x$

$14.540$: Primitive of $\sinh a x$
$14.541$: Primitive of $x \sinh a x$
$14.542$: Primitive of $x^2 \sinh a x$
$14.543$: Primitive of $\dfrac {\sinh a x} x$
$14.544$: Primitive of $\dfrac {\sinh a x} {x^2}$
$14.545$: Primitive of $\dfrac 1 {\sinh a x}$
$14.546$: Primitive of $\dfrac x {\sinh a x}$
$14.547$: Primitive of $\sinh^2 a x$
$14.548$: Primitive of $x \sinh^2 a x$
$14.549$: Primitive of $\dfrac 1 {\sinh^2 a x}$
$14.550$: Primitive of $\sinh a x \sinh p x$
For $a = \pm p$ see $14.547$: Primitive of $\sinh^2 a x$.
$14.551$: Primitive of $\sinh a x \sin p x$
$14.552$: Primitive of $\sinh a x \cos p x$
$14.553$: Primitive of $\dfrac 1 {p + q \sinh a x}$
$14.554$: Primitive of $\dfrac 1 {\left({p + q \sinh a x}\right)^2}$
$14.555$: Primitive of $\dfrac 1 {p^2 + q^2 \sinh^2 a x}$
$14.556$: Primitive of $\dfrac 1 {p^2 - q^2 \sinh^2 a x}$
$14.557$: Primitive of $x^m \sinh a x$
$14.558$: Primitive of $\sinh^n a x$
$14.559$: Primitive of $\dfrac {\sinh a x} {x^n}$
$14.560$: Primitive of $\dfrac 1 {\sinh^n a x}$
$14.561$: Primitive of $\dfrac x {\sinh^n a x}$


Integrals Involving $\cosh a x$

$14.562$: Primitive of $\cosh a x$
$14.563$: Primitive of $x \cosh a x$
$14.564$: Primitive of $x^2 \cosh a x$
$14.565$: Primitive of $\dfrac {\cosh a x} x$
$14.566$: Primitive of $\dfrac {\cosh a x} {x^2}$
$14.567$: Primitive of $\dfrac 1 {\cosh a x}$
$14.568$: Primitive of $\dfrac x {\cosh a x}$
$14.569$: Primitive of $\cosh^2 a x$
$14.570$: Primitive of $x \cosh^2 a x$
$14.571$: Primitive of $\dfrac 1 {\cosh^2 a x}$
$14.572$: Primitive of $\cosh a x \cosh p x$
$14.573$: Primitive of $\cosh a x \sin p x$
$14.574$: Primitive of $\cosh a x \cos p x$
$14.575$: Primitive of $\dfrac 1 {\cosh a x + 1}$
$14.576$: Primitive of $\dfrac 1 {\cosh a x - 1}$
$14.577$: Primitive of $\dfrac x {\cosh a x + 1}$
$14.578$: Primitive of $\dfrac x {\cosh a x - 1}$
$14.579$: Primitive of $\dfrac 1 {\left({\cosh a x + 1}\right)^2}$
$14.580$: Primitive of $\dfrac 1 {\left({\cosh a x - 1}\right)^2}$
$14.581$: Primitive of $\dfrac 1 {p + q \cosh a x}$
$14.582$: Primitive of $\dfrac 1 {\left({p + q \cosh a x}\right)^2}$
$14.583$: Primitive of $\dfrac 1 {p^2 + q^2 \cosh^2 a x}$
$14.584$: Primitive of $\dfrac 1 {p^2 - q^2 \cosh^2 a x}$
$14.585$: Primitive of $x^m \cosh a x$
$14.586$: Primitive of $\cosh^n a x$
$14.587$: Primitive of $\dfrac {\cosh a x} {x^n}$
$14.588$: Primitive of $\dfrac 1 {\cosh^n a x}$
$14.589$: Primitive of $\dfrac x {\cosh^n a x}$


Integrals Involving $\sinh a x$ and $\cosh a x$

$14.590$: Primitive of $\sinh a x \cosh a x$
$14.591$: Primitive of $\sinh p x \cosh q x$
$14.592$: Primitive of $\sinh^n a x \cosh a x$
$14.593$: Primitive of $\cosh^n a x \sinh a x$
$14.594$: Primitive of $\sinh^2 a x \cosh^2 a x$
$14.595$: Primitive of $\dfrac 1 {\sinh a x \cosh a x}$
$14.596$: Primitive of $\dfrac 1 {\sinh^2 a x \cosh a x}$
$14.597$: Primitive of $\dfrac 1 {\sinh a x \cosh^2 a x}$
$14.598$: Primitive of $\dfrac 1 {\sinh^2 a x \cosh^2 a x}$
$14.599$: Primitive of $\dfrac {\sinh^2 a x} {\cosh a x}$
$14.600$: Primitive of $\dfrac {\cosh^2 a x} {\sinh a x}$
$14.601$: Primitive of $\dfrac 1 {\cosh a x \left({1 + \sinh a x}\right)}$
$14.602$: Primitive of $\dfrac 1 {\sinh a x \left({\sinh a x + 1}\right)}$
$14.603$: Primitive of $\dfrac 1 {\sinh a x \left({\sinh a x - 1}\right)}$


Integrals Involving $\tanh a x$

$14.604$: Primitive of $\tanh a x$
$14.605$: Primitive of $\tanh^2 a x$
$14.606$: Primitive of $\tanh^3 a x$
$14.607$: Primitive of $\tanh^n a x \operatorname{sech}^2 a x$
$14.608$: Primitive of $\dfrac {\operatorname{sech}^2 a x} {\tanh a x}$
$14.609$: Primitive of $\dfrac 1 {\tanh a x}$
$14.610$: Primitive of $x \tanh a x$
$14.611$: Primitive of $\dfrac {\tanh a x} x$
$14.612$: Primitive of $x \tanh^2 a x$
$14.613$: Primitive of $\dfrac 1 {p + q \tanh a x}$
$14.614$: Primitive of $\tanh^n a x$


Integrals Involving $\coth a x$

$14.615$: Primitive of $\coth a x$
$14.616$: Primitive of $\coth^2 a x$
$14.617$: Primitive of $\coth^3 a x$
$14.618$: Primitive of $\coth^n a x \operatorname{csch}^2 a x$
$14.619$: Primitive of $\dfrac {\operatorname{csch}^2 a x} {\coth a x}$
$14.620$: Primitive of $\dfrac 1 {\coth a x}$
$14.621$: Primitive of $x \coth a x$
$14.622$: Primitive of $\dfrac {\coth a x} x$
$14.623$: Primitive of $x \coth^2 a x$
$14.624$: Primitive of $\dfrac 1 {p + q \coth a x}$
$14.625$: Primitive of $\coth^n a x$


Integrals Involving $\operatorname{sech} a x$

$14.626$: Primitive of $\operatorname{sech} a x$
$14.627$: Primitive of $\operatorname{sech}^2 a x$
$14.628$: Primitive of $\operatorname{sech}^3 a x$
$14.629$: Primitive of $\operatorname{sech}^n a x \tanh a x$
$14.630$: Primitive of $\dfrac 1 {\operatorname{sech} a x}$
$14.631$: Primitive of $x \operatorname{sech} a x$
$14.632$: Primitive of $x \operatorname{sech}^2 a x$
$14.633$: Primitive of $\dfrac {\operatorname{sech} a x} x$
$14.634$: Primitive of $\dfrac 1 {q + p \operatorname{sech} a x}$
$14.635$: Primitive of $\operatorname{sech}^n a x$


Integrals Involving $\operatorname{csch} a x$

$14.636$: Primitive of $\operatorname{csch} a x$
$14.637$: Primitive of $\operatorname{csch}^2 a x$
$14.638$: Primitive of $\operatorname{csch}^3 a x$
$14.639$: Primitive of $\operatorname{csch}^n a x \coth a x$
$14.640$: Primitive of $\dfrac 1 {\operatorname{csch} a x}$
$14.641$: Primitive of $x \operatorname{csch} a x$
$14.642$: Primitive of $x \operatorname{csch}^2 a x$
$14.643$: Primitive of $\dfrac {\operatorname{csch} a x} x$
$14.644$: Primitive of $\dfrac 1 {q + p \operatorname{csch} a x}$
$14.645$: Primitive of $\operatorname{csch}^n a x$


Integrals Involving Inverse Hyperbolic Functions

$14.646$: Primitive of $\sinh^{-1} \dfrac x a$
$14.647$: Primitive of $x \sinh^{-1} \dfrac x a$
$14.648$: Primitive of $x^2 \sinh^{-1} \dfrac x a$
$14.649$: Primitive of $\dfrac {\sinh^{-1} \dfrac x a} x$
$14.650$: Primitive of $\dfrac {\sinh^{-1} \dfrac x a} {x^2}$
$14.651$: Primitive of $\cosh^{-1} \dfrac x a$
$14.652$: Primitive of $x \cosh^{-1} \dfrac x a$
$14.653$: Primitive of $x^2 \cosh^{-1} \dfrac x a$
$14.654$: Primitive of $\dfrac {\cosh^{-1} \dfrac x a} x$
$14.655$: Primitive of $\dfrac {\cosh^{-1} \dfrac x a} {x^2}$
$14.656$: Primitive of $\tanh^{-1} \dfrac x a$
$14.657$: Primitive of $x \tanh^{-1} \dfrac x a$
$14.658$: Primitive of $x^2 \tanh^{-1} \dfrac x a$
$14.659$: Primitive of $\dfrac {\tanh^{-1} \dfrac x a} x$
$14.660$: Primitive of $\dfrac {\tanh^{-1} \dfrac x a} {x^2}$
$14.661$: Primitive of $\coth^{-1} \dfrac x a$
$14.662$: Primitive of $x \coth^{-1} \dfrac x a$
$14.663$: Primitive of $x^2 \coth^{-1} \dfrac x a$
$14.664$: Primitive of $\dfrac {\coth^{-1} \dfrac x a} x$
$14.665$: Primitive of $\dfrac {\coth^{-1} \dfrac x a} {x^2}$
$14.666$: Primitive of $\operatorname{sech}^{-1} \dfrac x a$
$14.667$: Primitive of $x \operatorname{sech}^{-1} \dfrac x a$
$14.668$: Primitive of $\dfrac {\operatorname{sech}^{-1} \dfrac x a} x$
$14.669$: Primitive of $\operatorname{csch}^{-1} \dfrac x a$
$14.670$: Primitive of $x \operatorname{csch}^{-1} \dfrac x a$
$14.671$: Primitive of $\dfrac {\operatorname{csch}^{-1} \dfrac x a} x$
$14.672$: Primitive of $x^m \sinh^{-1} \dfrac x a$
$14.673$: Primitive of $x^m \cosh^{-1} \dfrac x a$
$14.674$: Primitive of $x^m \tanh^{-1} \dfrac x a$
$14.675$: Primitive of $x^m \coth^{-1} \dfrac x a$
$14.676$: Primitive of $x^m \operatorname{sech}^{-1} \dfrac x a$
$14.677$: Primitive of $x^m \operatorname{csch}^{-1} \dfrac x a$


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