What is the integral of xcos^2(x/2) ?

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

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integral of xcos^2(x/2)

\int x cos^{2} (\frac{x}{2}) d x

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

Solution steps

\int x cos^{2} (\frac{x}{2}) d x

Apply Integration By Parts

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

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

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

Add a constant to the solution

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

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