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 (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

= \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

2 y^{^{''}} + 16 y^{^{'}} + 32 y = 0, y (0) = - 3, y^{^{'}} (0) = 10

\frac{d}{d x} (f (x) (x)^{3} (x))

f (x) = cos (2 x)

\int \frac{1}{( 2 x + 1 ) ^{2}} d x

\int sec^{5} (x) tan^{3} (x) d x

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