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