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:
- $\ds \int \frac {x^{2 n - 1} \rd x} {\paren {a x^2 + b x + c}^n} = \frac 1 a \int \frac {x^{2 n - 3} \rd x} {\paren {a x^2 + b x + c}^{n - 1} } - \frac c a \int \frac {x^{2 n - 3} \rd x} {\paren {a x^2 + b x + c}^n} - \frac b a \int \frac {x^{2 n - 2} \rd x} {\paren {a x^2 + b x + c}^n}$
Proof
\(\ds \) | \(\) | \(\ds \int \frac {x^{2 n - 1} \rd x} {\paren {a x^2 + b x + c}^n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \int \frac {x^{2 n - 3} a x^2 \rd x} {\paren {a x^2 + b x + c}^n}\) | Primitive of Constant Multiple of Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \int \frac {x^{2 n - 3} \paren {a x^2 + b x + c - b x - c} \rd x} {\paren {a x^2 + b x + c}^n}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \int \frac {x^{2 n - 3} \rd x} {\paren {a x^2 + b x + c}^{n - 1} } - \frac c a \int \frac {x^{2 n - 3} \rd x} {\paren {a x^2 + b x + c}^n} - \frac b a \int \frac {x^{2 n - 2} \rd x} {\paren {a x^2 + b x + c}^n}\) | Linear Combination of Primitives |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $a x^2 + b x + c$: $14.276$