What is the integral of x*(4-x^2)^{-2/3} ?

The integral of x*(4-x^2)^{-2/3} is 3/10 x^2(4-x^2)^{2/3}-6/5 (-x^2+4)^{2/3}+C

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integral of x*(4-x^2)^{-2/3}

\int x \cdot (4 - x^{2})^{- \frac{2}{3}} d x

\frac{3}{10} x^{2} (4 - x^{2})^{\frac{2}{3}} - \frac{6}{5} (- x^{2} + 4)^{\frac{2}{3}} + C

Solution steps

\int x (4 - x^{2})^{- \frac{2}{3}} d x

- \frac{2}{3} = 2 \cdot \frac{1}{3}

= \int x (4 - x^{2})^{2 \cdot \frac{1}{3}} d x

Multiply

2 \cdot \frac{1}{3} : \frac{2}{3}

= \int x (4 - x^{2})^{\frac{2}{3}} d x

Apply Integration By Parts

: \frac{1}{2} x^{2} (4 - x^{2})^{\frac{2}{3}} - \int - \frac{2 x ^{3}}{3 ( - x ^{2} + 4 ) ^{\frac{1}{3}}} d x

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

\int - \frac{2 x ^{3}}{3 ( - x ^{2} + 4 ) ^{\frac{1}{3}}} d x = \frac{1}{5} (x^{2} + 6) (- x^{2} + 4)^{\frac{2}{3}}

= \frac{1}{2} x^{2} (4 - x^{2})^{\frac{2}{3}} - \frac{1}{5} (x^{2} + 6) (- x^{2} + 4)^{\frac{2}{3}}

Simplify

\frac{1}{2} x^{2} (4 - x^{2})^{\frac{2}{3}} - \frac{1}{5} (x^{2} + 6) (- x^{2} + 4)^{\frac{2}{3}} : \frac{3}{10} x^{2} (4 - x^{2})^{\frac{2}{3}} - \frac{6}{5} (- x^{2} + 4)^{\frac{2}{3}}

= \frac{3}{10} x^{2} (4 - x^{2})^{\frac{2}{3}} - \frac{6}{5} (- x^{2} + 4)^{\frac{2}{3}}

Add a constant to the solution

= \frac{3}{10} x^{2} (4 - x^{2})^{\frac{2}{3}} - \frac{6}{5} (- x^{2} + 4)^{\frac{2}{3}} + C

\int sec^{2} (x) + 8 d x

\int - ∣ x ∣ d x

\frac{d}{d x} (\frac{4 x ^{5} - 15}{x ^{4}})

y^{^{'}} = - \frac{y}{3}

\frac{d}{d x} (\frac{1}{1 + x ^{3}})

What is the integral of x*(4-x^2)^{-2/3} ?
The integral of x*(4-x^2)^{-2/3} is 3/10 x^2(4-x^2)^{2/3}-6/5 (-x^2+4)^{2/3}+C