What is the integral of cos(x)sqrt(sin(x)+2) ?

The integral of cos(x)sqrt(sin(x)+2) is 2/3 (sin(x)+2)^{3/2}+C

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integral of cos(x)sqrt(sin(x)+2)

\int cos (x) sin (x) + 2 d x

\frac{2}{3} (sin (x) + 2)^{\frac{3}{2}} + C

Solution steps

\int cos (x) sin (x) + 2 d x

Apply u-substitution

: \int u + 2 d u

= \int u + 2 d u

Apply u-substitution

: \int v d v

= \int v d v

Apply the Power Rule

: \frac{2}{3} v^{\frac{3}{2}}

= \frac{2}{3} v^{\frac{3}{2}}

Substitute back

= \frac{2}{3} (sin (x) + 2)^{\frac{3}{2}}

Add a constant to the solution

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

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What is the integral of cos(x)sqrt(sin(x)+2) ?
The integral of cos(x)sqrt(sin(x)+2) is 2/3 (sin(x)+2)^{3/2}+C