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

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

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integral of (1-x^2)/(xsqrt(x))

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

- \frac{2}{x} - \frac{2}{3} x^{\frac{3}{2}} + C

Solution steps

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

Expand

\frac{1 - x ^{2}}{x x} : \frac{1}{x x} - x

= \int \frac{1}{x x} - x d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{x x} d x - \int x d x

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

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

= - \frac{2}{x} - \frac{2}{3} x^{\frac{3}{2}}

Add a constant to the solution

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

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