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

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

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integral of xsin^3(x)

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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