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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

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