What is the integral of 4cos^5(x) ?

The integral of 4cos^5(x) is 4(sin(x)-(2sin^3(x))/3+(sin^5(x))/5)+C

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integral of 4cos^5(x)

\int 4 cos^{5} (x) d x

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

Solution steps

\int 4 cos^{5} (x) d x

Take the constant out:

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

= 4 \cdot \int cos^{5} (x) d x

Rewrite using trig identities

= 4 \cdot \int (1 - sin^{2} (x))^{2} cos (x) d x

Apply u-substitution

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

= 4 \cdot \int (1 - u^{2})^{2} d u

Expand

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

= 4 \cdot \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

= 4 (\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}

= 4 (u - \frac{2 u ^{3}}{3} + \frac{u ^{5}}{5})

Substitute back

u = sin (x)

= 4 (sin (x) - \frac{2 sin ^{3} ( x )}{3} + \frac{sin ^{5} ( x )}{5})

Add a constant to the solution

= 4 (sin (x) - \frac{2 sin ^{3} ( x )}{3} + \frac{sin ^{5} ( x )}{5}) + C

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What is the integral of 4cos^5(x) ?
The integral of 4cos^5(x) is 4(sin(x)-(2sin^3(x))/3+(sin^5(x))/5)+C