What is the integral of xarcsin(x/2) ?

The integral of xarcsin(x/2) is 1/2 x^2arcsin(x/2)-arcsin(1/2 x)+1/2 sin(2arcsin(1/2 x))+C

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integral of xarcsin(x/2)

\int x arcsin (\frac{x}{2}) d x

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

Solution steps

\int x arcsin (\frac{x}{2}) d x

Apply Integration By Parts

: \frac{1}{2} x^{2} arcsin (\frac{x}{2}) - \int \frac{x ^{2}}{2 4 - x ^{2}} d x

= \frac{1}{2} x^{2} arcsin (\frac{x}{2}) - \int \frac{x ^{2}}{2 4 - x ^{2}} d x

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

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

Simplify

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

Add a constant to the solution

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

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