What is the integral of x^3(1-x^2)^{-1/2} ?

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

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integral of x^3(1-x^2)^{-1/2}

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

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

Solution steps

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

Simplify

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

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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