What is the integral of x\sqrt[5]{5-x^2} ?

The integral of x\sqrt[5]{5-x^2} is -5/12 (5-x^2)^{6/5}+C

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integral of x\sqrt[5]{5-x^2}

- \frac{5}{12} (5 - x^{2})^{\frac{6}{5}} + C

Solution steps

Apply u-substitution

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

Apply the Power Rule

= - \frac{1}{2} \cdot \frac{5}{6} u^{\frac{6}{5}}

Substitute back

u = 5 - x^{2}

= - \frac{1}{2} \cdot \frac{5}{6} (5 - x^{2})^{\frac{6}{5}}

Simplify

- \frac{1}{2} \cdot \frac{5}{6} (5 - x^{2})^{\frac{6}{5}} : - \frac{5}{12} (5 - x^{2})^{\frac{6}{5}}

= - \frac{5}{12} (5 - x^{2})^{\frac{6}{5}}

Add a constant to the solution

= - \frac{5}{12} (5 - x^{2})^{\frac{6}{5}} + C

What is the integral of x\sqrt[5]{5-x^2} ?
The integral of x\sqrt[5]{5-x^2} is -5/12 (5-x^2)^{6/5}+C