What is the integral of xsqrt(1-x^2)arcsin(x) ?

The integral of xsqrt(1-x^2)arcsin(x) is -1/3 arcsin(x)(1-x^2)^{3/2}+1/3 (x-(x^3)/3)+C

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integral of xsqrt(1-x^2)arcsin(x)

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

= - \frac{1}{3} arcsin (x) (1 - x^{2})^{\frac{3}{2}} + \frac{1}{3} (x - \frac{x ^{3}}{3})

Add a constant to the solution

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

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

\int t^{3} e^{13 t} d t

\int \frac{2}{x ^{2} + 4 x + 5} d x

\int x d x

What is the integral of xsqrt(1-x^2)arcsin(x) ?
The integral of xsqrt(1-x^2)arcsin(x) is -1/3 arcsin(x)(1-x^2)^{3/2}+1/3 (x-(x^3)/3)+C