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

The integral of xtan^2(x/2) is -x^2+2xtan(x/2)+(x^2)/2+4ln|cos(x/2)|+C

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integral of xtan^2(x/2)

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

- x^{2} + 2 x tan (\frac{x}{2}) + \frac{x ^{2}}{2} + 4 ln cos (\frac{x}{2}) + C

Solution steps

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

Apply Integration By Parts

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

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

= x (- x + 2 tan (\frac{x}{2})) - (- \frac{x ^{2}}{2} - 4 ln cos (\frac{x}{2}))

Simplify

x (- x + 2 tan (\frac{x}{2})) - (- \frac{x ^{2}}{2} - 4 ln cos (\frac{x}{2})) : - x^{2} + 2 x tan (\frac{x}{2}) + \frac{x ^{2}}{2} + 4 ln cos (\frac{x}{2})

= - x^{2} + 2 x tan (\frac{x}{2}) + \frac{x ^{2}}{2} + 4 ln cos (\frac{x}{2})

Add a constant to the solution

= - x^{2} + 2 x tan (\frac{x}{2}) + \frac{x ^{2}}{2} + 4 ln cos (\frac{x}{2}) + C

What is the integral of xtan^2(x/2) ?
The integral of xtan^2(x/2) is -x^2+2xtan(x/2)+(x^2)/2+4ln|cos(x/2)|+C