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

The integral of x(cos(x))^3 is xsin(x)-1/3 xsin^3(x)+2/3 cos(x)+1/9 cos^3(x)+C

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integral of x(cos(x))^3

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

x sin (x) - \frac{1}{3} x sin^{3} (x) + \frac{2}{3} cos (x) + \frac{1}{9} cos^{3} (x) + C

Solution steps

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

Simplify

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

Apply Integration By Parts

: x (sin (x) - \frac{1}{3} sin^{3} (x)) - \int sin (x) - \frac{1}{3} sin^{3} (x) d x

= x (sin (x) - \frac{1}{3} sin^{3} (x)) - \int sin (x) - \frac{1}{3} sin^{3} (x) d x

\int sin (x) - \frac{1}{3} sin^{3} (x) d x = - \frac{2}{3} cos (x) - \frac{1}{9} cos^{3} (x)

= x (sin (x) - \frac{1}{3} sin^{3} (x)) - (- \frac{2}{3} cos (x) - \frac{1}{9} cos^{3} (x))

Simplify

x (sin (x) - \frac{1}{3} sin^{3} (x)) - (- \frac{2}{3} cos (x) - \frac{1}{9} cos^{3} (x)) : x sin (x) - \frac{1}{3} x sin^{3} (x) + \frac{2}{3} cos (x) + \frac{1}{9} cos^{3} (x)

= x sin (x) - \frac{1}{3} x sin^{3} (x) + \frac{2}{3} cos (x) + \frac{1}{9} cos^{3} (x)

Add a constant to the solution

= x sin (x) - \frac{1}{3} x sin^{3} (x) + \frac{2}{3} cos (x) + \frac{1}{9} cos^{3} (x) + C

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What is the integral of x(cos(x))^3 ?
The integral of x(cos(x))^3 is xsin(x)-1/3 xsin^3(x)+2/3 cos(x)+1/9 cos^3(x)+C