Primitive of x over a x + b by p x + q

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Theorem

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


Proof

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

$\blacksquare$


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