Primitive of Reciprocal of x by a x + b squared

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Theorem

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


Proof 1

\(\displaystyle \int \frac {\mathrm d x} {x \left({a x + b}\right)^2}\) \(=\) \(\displaystyle \int \left({\frac 1 {b^2 x} - \frac a {b^2 \left({a x + b}\right)} - \frac a {b \left({a x + b}\right)^2} }\right) \ \mathrm d x\) Partial Fraction Expansion
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {b^2} \int \frac {\mathrm d x} x - \frac a {b^2} \int \frac {\mathrm d x} {a x + b} - \frac a b \int \frac {\mathrm d x} {\left({a x + b}\right)^2}\) Linear Combination of Integrals
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {b^2} \ln \left\vert{x}\right\vert - \frac a {b^2} \int \frac {\mathrm d x} {a x + b} - \frac a b \int \frac {\mathrm d x} {\left({a x + b}\right)^2} + C\) Primitive of Reciprocal
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {b^2} \ln \left\vert{x}\right\vert - \frac a {b^2} \ln \left\vert{a x + b}\right\vert - \frac a b \int \frac {\mathrm d x} {\left({a x + b}\right)^2} + C\) Primitive of $\dfrac 1 {a x + b}$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {b^2} \ln \left\vert{x}\right\vert - \frac a {b^2} \ln \left\vert{a x + b}\right\vert - \frac a b \frac {-1} {a \left({a x + b}\right)} + C\) Primitive of $\dfrac 1 {\left({a x + b}\right)^2}$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {b \left({a x + b}\right)} + \frac 1 {b^2} \ln \left\vert{\frac x {a x + b} }\right\vert + C\) Difference of Logarithms and rearranging

$\blacksquare$


Proof 2

\(\displaystyle \int \frac {\mathrm d x} {x \left({a x + b}\right)^2}\) \(=\) \(\displaystyle \int \frac {b \ \mathrm d x} {b x \left({a x + b}\right)^2}\) multiplying top and bottom by $b$
\(\displaystyle \) \(=\) \(\displaystyle \int \frac {\left({a x + b - a x}\right) \ \mathrm d x} {b x \left({a x + b}\right)^2}\) adding and subtracting $a x$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 b \int \frac {\left({a x + b}\right) \ \mathrm d x} {x \left({a x + b}\right)^2} - \frac a b \int \frac {x \ \mathrm d x} {x \left({a x + b}\right)^2}\) Linear Combination of Integrals
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 b \int \frac {\mathrm d x} {x \left({a x + b}\right)} - \frac a b \int \frac {\mathrm d x} {\left({a x + b}\right)^2}\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 b \left({\frac 1 b \ln \left\vert{\frac x {a x + b} }\right\vert}\right) - \frac a b \int \frac {\mathrm d x} {\left({a x + b}\right)^2} + C\) Primitive of $\dfrac 1 {x \left({a x + b}\right)}$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {b^2} \ln \left\vert{\frac x {a x + b} }\right\vert - \frac a b \left({-\frac 1 {a \left({a x + b}\right)} }\right) + C\) Primitive of $\dfrac 1 {\left({a x + b}\right)^2}$
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {b \left({a x + b}\right)} + \frac 1 {b^2} \ln \left\vert{\frac x {a x + b} }\right\vert + C\) simplifying

$\blacksquare$


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