What is the integral of cos(x)cos^3(sin(x)) ?

The integral of cos(x)cos^3(sin(x)) is sin(sin(x))-(sin^3(sin(x)))/3+C

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integral of cos(x)cos^3(sin(x))

\int cos (x) cos^{3} (sin (x)) d x

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

Solution steps

\int cos (x) cos^{3} (sin (x)) d x

Apply u-substitution

= \int cos^{3} (u) d u

Rewrite using trig identities

= \int (1 - sin^{2} (u)) cos (u) d u

Apply u-substitution

= \int 1 - v^{2} d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 1 d v - \int v^{2} d v

\int 1 d v = v

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

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

Substitute back

= sin (sin (x)) - \frac{sin ^{3} ( sin ( x ) )}{3}

Add a constant to the solution

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

What is the integral of cos(x)cos^3(sin(x)) ?
The integral of cos(x)cos^3(sin(x)) is sin(sin(x))-(sin^3(sin(x)))/3+C