Primitive of x by Half Integer Power of a x squared plus b x plus c
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Theorem
Let $a \in \R_{\ne 0}$.
Then:
- $\ds \int x \paren {a x^2 + b x + c}^{n + \frac 1 2} \rd x = \frac {\paren {a x^2 + b x + c}^{n + \frac 3 2} } {a \paren {2 n + 3} } - \frac b {2 a} \int \paren {a x^2 + b x + c}^{n + \frac 1 2} \rd x$
Proof
\(\ds \) | \(\) | \(\ds \int x \paren {a x^2 + b x + c}^{n + \frac 1 2} \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {2 a x \paren {a x^2 + b x + c}^{n + \frac 1 2} \rd x} {2 a}\) | multiplying top and bottom by $2 a$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \int \frac {\paren {2 a x + b - b} \paren {a x^2 + b x + c}^{n + \frac 1 2} \rd x} {2 a}\) | ||||||||||||
\(\text {(1)}: \quad\) | \(\ds \) | \(=\) | \(\ds \frac 1 {2 a} \int \paren {2 a x + b} \paren {a x^2 + b x + c}^{n + \frac 1 2} \rd x - \frac b {2 a} \int \paren {a x^2 + b x + c}^{n + \frac 1 2} \rd x\) | Linear Combination of Primitives |
Let:
\(\ds z\) | \(=\) | \(\ds a x^2 + b x + c\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d z} {\d x}\) | \(=\) | \(\ds 2 a x + b\) | Derivative of Power | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \paren {2 a x + b} \paren {a x^2 + b x + c}^{n + \frac 1 2} \rd x\) | \(=\) | \(\ds \int z^{n + \frac 1 2} \rd z\) | Integration by Substitution | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac {z^{n + \frac 3 2} } {n + \frac 3 2}\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 z^{n + \frac 3 2} } {2 n + 3}\) | simplifying | |||||||||||
\(\text {(2)}: \quad\) | \(\ds \) | \(=\) | \(\ds \frac {2 \paren {a x^2 + b x + c}^{n + \frac 3 2} } {2 n + 3}\) | substituting for $z$ |
So:
\(\ds \) | \(\) | \(\ds \int x \paren {a x^2 + b x + c}^{n + \frac 1 2} \rd x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a} \int \paren {2 a x + b} \paren {a x^2 + b x + c}^{n + \frac 1 2} \rd x - \frac b {2 a} \int \paren {a x^2 + b x + c}^{n + \frac 1 2} \rd x\) | from $(1)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {2 a} \paren {\frac {2 \paren {a x^2 + b x + c}^{n + \frac 3 2} } {2 n + 3} } - \frac b {2 a} \int \paren {a x^2 + b x + c}^{n + \frac 1 2} \rd x\) | from $(2)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {a x^2 + b x + c}^{n + \frac 3 2} } {a \paren {2 n + 3} } - \frac b {2 a} \int \paren {a x^2 + b x + c}^{n + \frac 1 2} \rd x\) | simplifying |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sqrt {a x^2 + b x + c}$: $14.296$