How do you integrate $\int cos x \sqrt{sin x}$ $int cosxsqrtsinx$ using substitution?

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How do you integrate $\int cos x \sqrt{sin x}$ $int cosxsqrtsinx$ using substitution?

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Answer 1 · 2016-12-18T14:40:48

Let $u = sin x$ $u = sinx$ . Then $d u = cos x d x$ $du = cosxdx$ and $d x = \frac{d u}{cos x}$ $dx = (du)/cosx$ .

$= \int \frac{cos x \sqrt{u}}{cos x} d u$ $=int(cosxsqrt(u))/cosx du$

$= \sqrt{u} d u$ $=sqrt(u) du$

$= u^{\frac{1}{2}} d u$ $= u^(1/2) du$

$= \frac{2}{3} u^{\frac{3}{2}} + C$ $=2/3u^(3/2) + C$

$= \frac{2}{3} {(sin x)}^{\frac{3}{2}} + C$ $=2/3(sinx)^(3/2) + C$

$= \frac{2}{3} \sqrt{{sin}^{3} x} + C$ $=2/3sqrt(sin^3x) + C$

$= \frac{2}{3} sin x \sqrt{sin x} + C$ $=2/3sinxsqrt(sinx) + C$

Hopefully this helps!

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How do you integrate $\int cos x \sqrt{sin x}$ $int cosxsqrtsinx$ using substitution?

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