What is the integral of arcsin(x)*x ?

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

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integral of arcsin(x)*x

\int arcsin (x) \cdot x d x

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

Solution steps

\int arcsin (x) x d x

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

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