What is the integral of cos^4(u) ?

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

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integral of cos^4(u)

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

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

Solution steps

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

Simplify

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

= \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

= 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))

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

Simplify

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

Add a constant to the solution

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

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