Primitive of Power of Root of a x + b
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Theorem
- $\ds \int \paren {\sqrt {a x + b} }^m \rd x = \frac {2 \paren {\sqrt {a x + b} }^{m + 2} } {a \paren {m + 2} } + C$
for $m \ne -2$.
Proof 1
Let $u = \sqrt {a x + b}$.
Then:
\(\ds \int \paren {\sqrt {a x + b} }^m \rd x\) | \(=\) | \(\ds \frac 2 a \int u \cdot u^m \rd x\) | Primitive of Function of $\sqrt {a x + b}$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 a \int u^{m + 1} \rd x\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 a \frac {u^{m + 2} } {m + 2} + C\) | Primitive of Power valid for $m + 2 \ne 0$, that is, $m \ne -2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 \paren {\sqrt {a x + b} }^{m + 2} } {a \paren {m + 2} } + C\) | substituting for $u$ |
$\blacksquare$
Proof 2
Let $u = a x + b$.
Then:
\(\ds \int \paren {\sqrt {a x + b} }^m \rd x\) | \(=\) | \(\ds \frac 1 a \int \paren {\sqrt u}^m \rd u\) | Primitive of Function of $a x + b$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \int u^{m / 2} \rd u\) | simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \frac {u^{m / 2 + 1} } {m / 2 + 1} + C\) | Primitive of Power valid for $m / 2 + 1 \ne 0$, that is, $m \ne -2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 a \frac {u^{\paren {m + 2} / 2} } {\paren {m + 2} / 2} + C\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 2 a \frac {\paren {\sqrt u}^{m + 2} } {\paren {m + 2} } + C\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 \paren {\sqrt {a x + b} }^{m + 2} } {a \paren {m + 2} } + C\) | substituting for $u$ |
$\blacksquare$
Also presented as
This result is also seen presented in the form:
- $\ds \int \paren {\sqrt {a x + b} }^n \rd x = \frac 2 a \frac {\paren {\sqrt {a x + b} }^{n + 2} } {n + 2} + C$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: Integrals involving $\sqrt {a x + b}$: $14.99$