Primitive of Reciprocal of x squared by a x + b/Proof 2

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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

\(\ds \int \frac {\d x} {x^2 \paren {a x + b} }\) \(=\) \(\ds \int \frac {b \rd x} {b x^2 \paren {a x + b} }\) 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

$\blacksquare$