What is the integral of 4cos^4(θ) ?

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

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integral of 4cos^4(θ)

\int 4 cos^{4} (θ) d θ

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

Solution steps

\int 4 cos^{4} (θ) d θ

Take the constant out:

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

= 4 \cdot \int cos^{4} (θ) d θ

Simplify

cos^{4} (θ) : cos^{3} (θ) cos (θ)

= 4 \cdot \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

= 4 (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))

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

Simplify

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

Add a constant to the solution

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