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

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Theorem

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


Proof

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

$\blacksquare$


Sources

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