Primitive of x squared by Power of Root of a x + b
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Theorem
- $\ds \int x^2 \paren {\sqrt {a x + b} }^m \rd x = \frac {2 \paren {\sqrt {a x + b} }^{m + 6} } {a^3 \paren {m + 6} } - \frac {4 b \paren {\sqrt {a x + b} }^{m + 4} } {a^3 \paren {m + 4} } + \frac {2 b^2 \paren {\sqrt {a x + b} }^{m + 2} } {a^3 \paren {m + 2} } + C$
Proof
Let:
\(\ds u\) | \(=\) | \(\ds \sqrt {a x + b}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds \frac {u^2 - b} a\) |
Then:
\(\ds \int x^2 \paren {\sqrt {a x + b} }^m \rd x\) | \(=\) | \(\ds \frac 2 a \int \paren {\frac {u^2 - b} a}^2 u^{m + 1} \rd x\) | Primitive of Function of $a x + b$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 {a^3} \int u^{m + 5} \rd x - \frac {4 b} {a^3} \int u^{m + 3} \rd x + \frac {2 b^2} {a^3} \int u^{m + 1} \rd x\) | Linear Combination of Primitives | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 {a^3} \frac {u^{m + 6} } {m + 6} - \frac {4 b} {a^3} \frac {u^{m + 4} } {m + 4} + \frac {2 b^2} {a^3} \frac {u^{m + 2} } {m + 2} + C\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 \paren {\sqrt {a x + b} }^{m + 6} } {a^3 \paren {m + 6} } - \frac {4 b \paren {\sqrt {a x + b} }^{m + 4} } {a^3 \paren {m + 4} } + \frac {2 b^2 \paren {\sqrt {a x + b} }^{m + 2} } {a^3 \paren {m + 2} } + C\) | substituting for $u$ |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sqrt {a x + b}$: $14.101$