What is the integral of x(cos^2(x)) ?

The integral of x(cos^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(cos^2(x))

\int x (cos^{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 cos^{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

y = x^{2} - 2 x - 8

\int (x^{2} + y) d x

\frac{d}{d y} (\frac{1}{y ^{2} + 1})

y^{^{'}} + 2 x y = 2 x^{3}

\frac{d}{d x} (\frac{x ^{2} - 3}{x + 2})

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