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

The integral of cos(x)sin(x)cos(sin^2(x)) is 1/2 sin(sin^2(x))+C

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integral of cos(x)sin(x)cos(sin^2(x))

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

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

Solution steps

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

Rewrite using trig identities

= \int \frac{1}{2} sin (2 x) cos (sin^{2} (x)) d x

Take the constant out:

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

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

Apply u-substitution

: \int cos (u) d u

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

Use the common integral:

\int cos (u) d u = sin (u)

= \frac{1}{2} sin (u)

Substitute back

u = sin^{2} (x)

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

Add a constant to the solution

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

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What is the integral of cos(x)sin(x)cos(sin^2(x)) ?
The integral of cos(x)sin(x)cos(sin^2(x)) is 1/2 sin(sin^2(x))+C