What is the integral of 3x^2*arcsin(x) ?

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

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integral of 3x^2*arcsin(x)

\int 3 x^{2} \cdot arcsin (x) d x

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

Solution steps

\int 3 x^{2} arcsin (x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \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

= 3 (\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})

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

Simplify

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

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

Add a constant to the solution

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

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