What is the integral of xarcsin(x/3) ?

The integral of xarcsin(x/3) is 1/2 x^2arcsin(x/3)-9/4 (arcsin(1/3 x)-1/2 sin(2arcsin(1/3 x)))+C

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integral of xarcsin(x/3)

\int x arcsin (\frac{x}{3}) d x

\frac{1}{2} x^{2} arcsin (\frac{x}{3}) - \frac{9}{4} (arcsin (\frac{1}{3} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{3} x))) + C

Solution steps

\int x arcsin (\frac{x}{3}) d x

Apply Integration By Parts

= \frac{1}{2} x^{2} arcsin (\frac{x}{3}) - \int \frac{x ^{2}}{2 9 - x ^{2}} d x

\int \frac{x ^{2}}{2 9 - x ^{2}} d x = \frac{9}{4} (arcsin (\frac{1}{3} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{3} x)))

= \frac{1}{2} x^{2} arcsin (\frac{x}{3}) - \frac{9}{4} (arcsin (\frac{1}{3} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{3} x)))

Add a constant to the solution

= \frac{1}{2} x^{2} arcsin (\frac{x}{3}) - \frac{9}{4} (arcsin (\frac{1}{3} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{3} x))) + C

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What is the integral of xarcsin(x/3) ?
The integral of xarcsin(x/3) is 1/2 x^2arcsin(x/3)-9/4 (arcsin(1/3 x)-1/2 sin(2arcsin(1/3 x)))+C