What is the integral of (sqrt(1-x^3))/(x^2*sqrt(x)) ?

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

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integral of (sqrt(1-x^3))/(x^2*sqrt(x))

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

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