# Primitive of Reciprocal of x squared by a x + b

## Theorem

$\ds \int \frac {\d x} {x^2 \paren {a x + b} } = -\frac 1 {b x} + \frac a {b^2} \ln \size {\frac {a x + b} x} + C$

## Proof 1

 $\ds \int \frac {\d x} {x^2 \paren {a x + b} }$ $=$ $\ds \int \paren {-\frac a {b^2 x} + \frac 1 {b x^2} + \frac {a^2} {b^2 \paren {a x + b} } } \rd x$ Partial Fraction Expansion $\ds$ $=$ $\ds -\frac a {b^2} \int \frac {\d x} x + \frac 1 b \int \frac {\d x} {x^2} + \frac {a^2} {b^2} \int \frac {\d x} {a x + b}$ Linear Combination of Primitives $\ds$ $=$ $\ds -\frac a {b^2} \int \frac {\d x} x + \frac 1 b \frac {-1} x + \frac {a^2} {b^2} \int \frac {\d x} {a x + b} + C$ Primitive of Power $\ds$ $=$ $\ds -\frac a {b^2} \ln \size x - \frac 1 {b x} + \frac {a^2} {b^2} \int \frac {\d x} {a x + b} + C$ Primitive of Reciprocal $\ds$ $=$ $\ds -\frac a {b^2} \ln \size x - \frac 1 {b x} + \frac a {b^2} \ln \size {a x + b} + C$ Primitive of $\dfrac 1 {a x + b}$ $\ds$ $=$ $\ds -\frac 1 {b x} + \frac a {b^2} \ln \size {\frac {a x + b} x} + C$ Difference of Logarithms

$\blacksquare$

## Proof 2

 $\ds \int \frac {\d x} {x^2 \paren {a x + b} }$ $=$ $\ds \int \frac {b \rd x} {b x^2 \paren {a x + b} }$ multiplying top and bottom by $b$ $\ds$ $=$ $\ds \int \frac {\paren {a x + b - a x} \rd x} {b x^2 \paren {a x + b} }$ adding and subtracting $a x$ $\ds$ $=$ $\ds \frac 1 b \int \frac {\paren {a x + b } \rd x} {x^2 \paren {a x + b} } - \frac a b \int \frac {x \rd x} {x^2 \paren {a x + b} }$ Linear Combination of Primitives $\ds$ $=$ $\ds \frac 1 b \int \frac {\d x} {x^2} - \frac a b \int \frac {\d x} {x \paren {a x + b} }$ simplification $\ds$ $=$ $\ds -\frac 1 {b x} - \frac a b \int \frac {\d x} {x \paren {a x + b} } + C$ Primitive of Power $\ds$ $=$ $\ds -\frac 1 {b x} - \frac a b \paren {\frac 1 b \ln \size {\frac x {a x + b} } } + C$ Primitive of $\dfrac 1 {x \paren {a x + b} }$ $\ds$ $=$ $\ds -\frac 1 {b x} + \frac a {b^2} \ln \size {\frac {a x + b} x} + C$ Logarithm of Reciprocal and simplification

$\blacksquare$