What is the integral of xcos(x^2+4)sqrt(sin(x^2+4)) ?

The integral of xcos(x^2+4)sqrt(sin(x^2+4)) is 1/3 sin^{3/2}(x^2+4)+C

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integral of xcos(x^2+4)sqrt(sin(x^2+4))

\int x cos (x^{2} + 4) sin (x^{2} + 4) d x

\frac{1}{3} sin^{\frac{3}{2}} (x^{2} + 4) + C

Solution steps

\int x cos (x^{2} + 4) sin (x^{2} + 4) d x

Apply u-substitution

: \int \frac{cos ( u ) sin ( u )}{2} d u

= \int \frac{cos ( u ) sin ( u )}{2} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int cos (u) sin (u) d u

Apply u-substitution

: \int v d v

= \frac{1}{2} \cdot \int v d v

Apply the Power Rule

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

= \frac{1}{2} \cdot \frac{2}{3} v^{\frac{3}{2}}

Substitute back

= \frac{1}{2} \cdot \frac{2}{3} sin^{\frac{3}{2}} (x^{2} + 4)

Simplify

\frac{1}{2} \cdot \frac{2}{3} sin^{\frac{3}{2}} (x^{2} + 4) : \frac{1}{3} sin^{\frac{3}{2}} (x^{2} + 4)

= \frac{1}{3} sin^{\frac{3}{2}} (x^{2} + 4)

Add a constant to the solution

= \frac{1}{3} sin^{\frac{3}{2}} (x^{2} + 4) + C

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What is the integral of xcos(x^2+4)sqrt(sin(x^2+4)) ?
The integral of xcos(x^2+4)sqrt(sin(x^2+4)) is 1/3 sin^{3/2}(x^2+4)+C