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

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 $\ln \left\vert{u}\right\vert$]; all angles are in radians; all constants of integration are omitted but implied.


 * $14.1$: Primitive of $a$


 * $14.2$: Primitive of $a f \left({x}\right)$


 * $14.3$: Linear Combination of Integrals: $\displaystyle \int \left({u \pm v \pm w \pm \cdots}\right) \ \mathrm d x$


 * $14.4$: Integration by Parts: $\displaystyle \int u \ \mathrm d v$


 * $14.5$: Primitive of $f \left({a x}\right)$


 * $14.6$: Primitive of $F \left\{ {f \left({x}\right)}\right\}$


 * $14.7$: Primitive of $u^n$


 * $14.8$: Primitive of $\dfrac {\mathrm d u} u$


 * $14.9$: 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 $\operatorname{sech} u$


 * $14.30$: Primitive of $\operatorname{csch} u$


 * $14.31$: Primitive of $\operatorname{sech}^2 u$


 * $14.32$: Primitive of $\operatorname{csch}^2 u$


 * $14.33$: 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: $\displaystyle \int f^{\left({n}\right)}g \ \mathrm d x$