What is the integral of (cos(θ))^4 ?

The integral of (cos(θ))^4 is cos^3(θ)sin(θ)+3/8 (θ-1/4 sin(4θ))+C

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integral of (cos(θ))^4

\int (cos (θ))^{4} d θ

cos^{3} (θ) sin (θ) + \frac{3}{8} (θ - \frac{1}{4} sin (4 θ)) + C

Solution steps

\int (cos (θ))^{4} d θ

Simplify

= \int cos^{4} (θ) d θ

Simplify

cos^{4} (θ) : cos^{3} (θ) cos (θ)

= \int cos^{3} (θ) cos (θ) d θ

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

= cos^{3} (θ) sin (θ) + \frac{3}{8} (θ - \frac{1}{4} sin (4 θ)) + C

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What is the integral of (cos(θ))^4 ?
The integral of (cos(θ))^4 is cos^3(θ)sin(θ)+3/8 (θ-1/4 sin(4θ))+C