What is the integral of cos^4(x/3) ?

The integral of cos^4(x/3) is 3(cos^3(x/3)sin(x/3)+x/8-3/32 sin((4x)/3))+C

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integral of cos^4(x/3)

\int cos^{4} (\frac{x}{3}) d x

3 (cos^{3} (\frac{x}{3}) sin (\frac{x}{3}) + \frac{x}{8} - \frac{3}{32} sin (\frac{4 x}{3})) + C

Solution steps

\int cos^{4} (\frac{x}{3}) d x

Apply u-substitution

: \int cos^{4} (u) \cdot 3 d u

= \int cos^{4} (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^{4} (u) d u

Simplify

cos^{4} (u) : cos^{3} (u) cos (u)

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

Apply Integration By Parts

: cos^{3} (u) sin (u) - \int - 3 cos^{2} (u) sin^{2} (u) d u

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

\int - 3 cos^{2} (u) sin^{2} (u) d u = - \frac{3}{8} (u - \frac{1}{4} sin (4 u))

= 3 (cos^{3} (u) sin (u) - (- \frac{3}{8} (u - \frac{1}{4} sin (4 u))))

Substitute back

u = \frac{x}{3}

= 3 (cos^{3} (\frac{x}{3}) sin (\frac{x}{3}) - (- \frac{3}{8} (\frac{x}{3} - \frac{1}{4} sin (4 \cdot \frac{x}{3}))))

Simplify

3 (cos^{3} (\frac{x}{3}) sin (\frac{x}{3}) - (- \frac{3}{8} (\frac{x}{3} - \frac{1}{4} sin (4 \cdot \frac{x}{3})))) : 3 (cos^{3} (\frac{x}{3}) sin (\frac{x}{3}) + \frac{x}{8} - \frac{3}{32} sin (\frac{4 x}{3}))

= 3 (cos^{3} (\frac{x}{3}) sin (\frac{x}{3}) + \frac{x}{8} - \frac{3}{32} sin (\frac{4 x}{3}))

Add a constant to the solution

= 3 (cos^{3} (\frac{x}{3}) sin (\frac{x}{3}) + \frac{x}{8} - \frac{3}{32} sin (\frac{4 x}{3})) + C

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What is the integral of cos^4(x/3) ?
The integral of cos^4(x/3) is 3(cos^3(x/3)sin(x/3)+x/8-3/32 sin((4x)/3))+C