What is the integral of xarcsin(x) ?

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

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integral of xarcsin(x)

\int x arcsin (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 x arcsin (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

\frac{d}{d x} ((cos (x))^{- 1})

\int (- 14 sin (x)) d x

\frac{d}{d x} (ln (3 x^{2} + 2))

\int_{0}^{\infty} e^{- 0.4 x} d x

\int e^{x} + e^{- x} d x

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