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

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

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integral of cos^4(w)

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

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

Solution steps

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

Simplify

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

= \int cos^{3} (w) cos (w) d w

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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