Primitive of a x + b over p x + q

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Theorem

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


Proof

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

$\blacksquare$


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