Primitive of Square Root
(Redirected from Primitive of Power/Examples/Square Root of x)
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Theorem
- $\ds \int \sqrt x \rd x = \dfrac {2 x^{3 / 2} } 3 = \dfrac {2 \sqrt x^3} 3$
Proof
From Primitive of Power:
Let $n \in \R: n \ne -1$.
Then:
- $\ds \int x^n \rd x = \frac {x^{n + 1} } {n + 1} + C$
where $C$ is an arbitrary constant.
Hence:
\(\ds \int \sqrt x \rd x\) | \(=\) | \(\ds \int x^{1/2} \rd x\) | Definition of Square Root | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {x^{1/2 + 1} } {1/2 + 1} + C\) | Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {x^{3/2} } {3/2} + C\) | simplification | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {2 \sqrt x^3} 3 + C\) | simplification |
$\blacksquare$