Primitive of p x + q over a x squared plus 2 b x plus c

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Theorem

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


Proof

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

$\blacksquare$


Examples

Primitive of $\dfrac {2 x + 1} {3 x^2 + 4 x + 2}$

$\ds \int \dfrac {2 x + 1} {3 x^2 + 4 x + 2} \rd x = \dfrac 1 3 \ln \size {3 x^2 + 4 x + 2} - \dfrac 1 {3 \sqrt 2} \map \arctan {\dfrac {3 x + 2} {\sqrt 2} } + C$


Sources

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