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

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

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

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

\frac{x ^{3}}{( 1 - x ^{2} ) ^{\frac{1}{2}}} + x^{3} - x^{2} + 1 + \frac{1}{2} arcsin (x) + \frac{3}{4} sin (2 arcsin (x)) - \frac{2 x}{( - x ^{2} + 1 ) ^{\frac{1}{2}}} + \frac{x ^{5}}{( - x ^{2} + 1 ) ^{\frac{1}{2}}} + C

Solution steps

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

Apply Integration By Parts

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

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

\int \frac{x ( 2 x - x ^{3} )}{( 1 - x ^{2} ) ^{\frac{3}{2}}} d x = - x^{3} - x^{2} + 1 - \frac{1}{2} arcsin (x) - \frac{3}{4} sin (2 arcsin (x)) + \frac{2 x}{( - x ^{2} + 1 ) ^{\frac{1}{2}}} - \frac{x ^{5}}{( - x ^{2} + 1 ) ^{\frac{1}{2}}}

= \frac{x ^{3}}{( 1 - x ^{2} ) ^{\frac{1}{2}}} - (- x^{3} - x^{2} + 1 - \frac{1}{2} arcsin (x) - \frac{3}{4} sin (2 arcsin (x)) + \frac{2 x}{( - x ^{2} + 1 ) ^{\frac{1}{2}}} - \frac{x ^{5}}{( - x ^{2} + 1 ) ^{\frac{1}{2}}})

Simplify

= \frac{x ^{3}}{( 1 - x ^{2} ) ^{\frac{1}{2}}} + x^{3} - x^{2} + 1 + \frac{1}{2} arcsin (x) + \frac{3}{4} sin (2 arcsin (x)) - \frac{2 x}{( - x ^{2} + 1 ) ^{\frac{1}{2}}} + \frac{x ^{5}}{( - x ^{2} + 1 ) ^{\frac{1}{2}}}

Add a constant to the solution

= \frac{x ^{3}}{( 1 - x ^{2} ) ^{\frac{1}{2}}} + x^{3} - x^{2} + 1 + \frac{1}{2} arcsin (x) + \frac{3}{4} sin (2 arcsin (x)) - \frac{2 x}{( - x ^{2} + 1 ) ^{\frac{1}{2}}} + \frac{x ^{5}}{( - x ^{2} + 1 ) ^{\frac{1}{2}}} + C

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