What is the integral of (cos^5(x)) ?

The integral of (cos^5(x)) is sin(x)-(2sin^3(x))/3+(sin^5(x))/5+C

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integral of (cos^5(x))

\int (cos^{5} (x)) d x

sin (x) - \frac{2 sin ^{3} ( x )}{3} + \frac{sin ^{5} ( x )}{5} + C

Solution steps

\int cos^{5} (x) d x

Rewrite using trig identities

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

Apply u-substitution

: \int (1 - u^{2})^{2} d u

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

Expand

(1 - u^{2})^{2} : 1 - 2 u^{2} + u^{4}

= \int 1 - 2 u^{2} + u^{4} d u

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 u - \int 2 u^{2} d u + \int u^{4} d u

\int 1 d u = u

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

\int u^{4} d u = \frac{u ^{5}}{5}

= u - \frac{2 u ^{3}}{3} + \frac{u ^{5}}{5}

Substitute back

u = sin (x)

= sin (x) - \frac{2 sin ^{3} ( x )}{3} + \frac{sin ^{5} ( x )}{5}

Add a constant to the solution

= sin (x) - \frac{2 sin ^{3} ( x )}{3} + \frac{sin ^{5} ( x )}{5} + C

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What is the integral of (cos^5(x)) ?
The integral of (cos^5(x)) is sin(x)-(2sin^3(x))/3+(sin^5(x))/5+C