Primitive of Odd Power of x over Power of a x squared plus b x plus c

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Theorem

Let $a \in \R_{\ne 0}$.

Then:

$\displaystyle \int \frac {x^{2 n - 1} \ \mathrm d x} {\left({a x^2 + b x + c}\right)^n} = \frac 1 a \int \frac {x^{2 n - 3} \ \mathrm d x} {\left({a x^2 + b x + c}\right)^{n - 1} } - \frac c a \int \frac {x^{2 n - 3} \ \mathrm d x} {\left({a x^2 + b x + c}\right)^n} - \frac b a \int \frac {x^{2 n - 2} \ \mathrm d x} {\left({a x^2 + b x + c}\right)^n}$


Proof

\(\displaystyle \) \(\) \(\displaystyle \int \frac {x^{2 n - 1} \ \mathrm d x} {\left({a x^2 + b x + c}\right)^n}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 a \int \frac {x^{2 n - 3} a x^2 \ \mathrm d x} {\left({a x^2 + b x + c}\right)^n}\) Primitive of Constant Multiple of Function
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 a \int \frac {x^{2 n - 3} \left({a x^2 + b x + c - b x - c}\right) \ \mathrm d x} {\left({a x^2 + b x + c}\right)^n}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 a \int \frac {x^{2 n - 3} \ \mathrm d x} {\left({a x^2 + b x + c}\right)^{n-1} } - \frac c a \int \frac {x^{2 n - 3} \ \mathrm d x} {\left({a x^2 + b x + c}\right)^n} - \frac b a \int \frac {x^{2 n - 2} \ \mathrm d x} {\left({a x^2 + b x + c}\right)^n}\) Linear Combination of Integrals

$\blacksquare$


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