What is the integral of cos^5(x/3) ?

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

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integral of cos^5(x/3)

\int cos^{5} (\frac{x}{3}) d x

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

Solution steps

\int cos^{5} (\frac{x}{3}) d x

Apply u-substitution

: \int cos^{5} (u) \cdot 3 d u

= \int cos^{5} (u) \cdot 3 d u

Take the constant out:

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

= 3 \cdot \int cos^{5} (u) d u

Rewrite using trig identities

= 3 \cdot \int (1 - sin^{2} (u))^{2} cos (u) d u

Apply u-substitution

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

= 3 \cdot \int (1 - v^{2})^{2} d v

Expand

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

= 3 \cdot \int 1 - 2 v^{2} + v^{4} 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

= 3 (\int 1 d v - \int 2 v^{2} d v + \int v^{4} d v)

\int 1 d v = v

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

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

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

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

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