What is the integral of x*sin^2(x) ?

The integral of x*sin^2(x) is 1/2 x(x-1/2 sin(2x))-1/2 ((x^2)/2+1/4 cos(2x))+C

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integral of x*sin^2(x)

\int x \cdot sin^{2} (x) d x

\frac{1}{2} x (x - \frac{1}{2} sin (2 x)) - \frac{1}{2} (\frac{x ^{2}}{2} + \frac{1}{4} cos (2 x)) + C

Solution steps

\int x sin^{2} (x) d x

Apply Integration By Parts

= \frac{1}{2} x (x - \frac{1}{2} sin (2 x)) - \int \frac{1}{2} (x - \frac{1}{2} sin (2 x)) d x

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

= \frac{1}{2} x (x - \frac{1}{2} sin (2 x)) - \frac{1}{2} (\frac{x ^{2}}{2} + \frac{1}{4} cos (2 x))

Add a constant to the solution

= \frac{1}{2} x (x - \frac{1}{2} sin (2 x)) - \frac{1}{2} (\frac{x ^{2}}{2} + \frac{1}{4} cos (2 x)) + C

What is the integral of x*sin^2(x) ?
The integral of x*sin^2(x) is 1/2 x(x-1/2 sin(2x))-1/2 ((x^2)/2+1/4 cos(2x))+C