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

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

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integral of cos^3(x)cos(x)

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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