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

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Theorem

$\ds \int \frac {\d x} {x^2 \paren {a x + b}^2} = \frac {-a} {b^2 \paren {a x + b} } - \frac 1 {b^2 x} + \frac {2 a} {b^3} \ln \size {\frac {a x + b} x} + C$


Proof

\(\ds \int \frac {\d x} {x^2 \paren {a x + b}^2}\) \(=\) \(\ds \int \paren {-\frac {2 a} {b^3 x} + \frac 1 {b^2 x^2} + \frac {2 a^2} {b^3 \paren {a x + b} } + \frac {a^2} {b^2 \paren {a x + b}^2} } \rd x\) Partial Fraction Expansion
\(\ds \) \(=\) \(\ds -\frac {2 a} {b^3} \int \frac {\d x} x + \frac 1 {b^2} \int \frac {\d x} {x^2} + \frac {2 a^2} {b^3} \int \frac {\d x} {\paren {a x + b} } + \frac {a^2} {b^2} \int \frac {\d x} {\paren {a x + b}^2}\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds -\frac {2 a} {b^3} \int \frac {\d x} x + \frac 1 {b^2} \frac {-1} x + \frac {2 a^2} {b^3} \int \frac {\d x} {\paren {a x + b} } + \frac {a^2} {b^2} \int \frac {\d x} {\paren {a x + b}^2} + C\) Primitive of Power
\(\ds \) \(=\) \(\ds -\frac {2 a} {b^3} \ln \size x + \frac 1 {b^2} \frac {-1} x + \frac {2 a^2} {b^3} \int \frac {\d x} {\paren {a x + b} } + \frac {a^2} {b^2} \int \frac {\d x} {\paren {a x + b}^2} + C\) Primitive of $\dfrac 1 x$
\(\ds \) \(=\) \(\ds -\frac {2 a} {b^3} \ln \size x - \frac 1 {b^2 x} + \frac {2 a^2} {a b^3} \ln \size {a x + b} + \frac {a^2} {b^2} \int \frac {\d x} {\paren {a x + b}^2} + C\) Primitive of $\dfrac 1 {a x + b}$
\(\ds \) \(=\) \(\ds -\frac {2 a} {b^3} \ln \size x - \frac 1 {b^2 x} + \frac {2 a} {b^3} \ln \size {a x + b} + \frac {a^2} {b^2} \frac {-1} {a \paren {a x + b} } + C\) Primitive of $\dfrac 1 {\paren {a x + b}^2}$
\(\ds \) \(=\) \(\ds \frac {-a} {b^2 \paren {a x + b} } - \frac 1 {b^2 x} + \frac {2 a} {b^3} \ln \size {\frac {a x + b} x} + C\) Difference of Logarithms and rearranging

$\blacksquare$


Sources

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