Primitives of Rational Functions involving a x + b squared
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Theorem
This page gathers together the primitives of some rational functions involving $\left({a x + b}\right)^2$.
Primitive of Reciprocal of $\left({a x + b}\right)^2$
- $\ds \int \frac {\d x} {\paren {a x + b}^2} = -\frac 1 {a \paren {a x + b} } + C$
Primitive of $x$ over $\left({a x + b}\right)^2$
- $\ds \int \frac {x \rd x} {\paren {a x + b}^2} = \frac b {a^2 \paren {a x + b} } + \frac 1 {a^2} \ln \size {a x + b} + C$
Primitive of $x^2$ over $\left({a x + b}\right)^2$
- $\ds \int \frac {x^2 \rd x} {\paren {a x + b}^2} = \frac {a x + b} {a^3} - \frac {b^2} {a^3 \paren {a x + b} } - \frac {2 b} {a^3} \ln \size {a x + b} + C$
Primitive of $x^3$ over $\left({a x + b}\right)^2$
- $\ds \int \frac {x^3 \rd x} {\paren {a x + b}^2} = \frac {\paren {a x + b}^2} {2 a^4} - \frac {3 b \paren {a x + b} } {a^4} + \frac {b^3} {a^4 \paren {a x + b} } + \frac {3 b^2} {a^4} \ln \size {a x + b} + C$
Primitive of Reciprocal of $x$ by $\left({a x + b}\right)^2$
- $\ds \int \frac {\d x} {x \paren {a x + b}^2} = \frac 1 {b \paren {a x + b} } + \frac 1 {b^2} \ln \size {\frac x {a x + b} } + C$
Primitive of Reciprocal of $x^2$ by $\left({a x + b}\right)^2$
- $\ds \int \frac {\d x} {x^2 \paren {a x + b}^2} = \frac {-a} {b^2 \paren {a x + b} } - \frac 1 {b^2 x} + \frac {2 a} {b^3} \ln \size {\frac {a x + b} x} + C$
Primitive of Reciprocal of $x^3$ by$\left({a x + b}\right)^2$
- $\ds \int \frac {\d x} {x^3 \paren {a x + b}^2} = - \frac {\paren {a x + b}^2} {2 b^4 x^2} + \frac {3 a \paren {a x + b} } {b^4 x} - \frac {a^3 x} {b^4 \paren {a x + b} } + \frac {3 a^2} {b^4} \ln \size {\frac x {a x + b} } + C$