What is the integral of x^2*arcsin(x) ?

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

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integral of x^2*arcsin(x)

\int x^{2} \cdot arcsin (x) d x

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

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