What is the integral of xcos^3(x)sin(x) ?

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

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integral of xcos^3(x)sin(x)

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

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

Solution steps

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

Apply Integration By Parts

: - \frac{1}{4} x cos^{4} (x) - \int - \frac{1}{4} cos^{4} (x) d x

= - \frac{1}{4} x cos^{4} (x) - \int - \frac{1}{4} cos^{4} (x) d x

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

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

Simplify

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

Add a constant to the solution

= - \frac{1}{4} x cos^{4} (x) + \frac{1}{4} (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 xcos^3(x)sin(x) ?
The integral of xcos^3(x)sin(x) is -1/4 xcos^4(x)+1/4 (cos^3(x)sin(x)+3/8 (x-1/4 sin(4x)))+C