What is the integral of (5x(2-x^2)^{1/4})/4 ?

The integral of (5x(2-x^2)^{1/4})/4 is -1/2 (2-x^2)^{5/4}+C

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integral of (5x(2-x^2)^{1/4})/4

\int \frac{5 x ( 2 - x ^{2} ) ^{\frac{1}{4}}}{4} d x

- \frac{1}{2} (2 - x^{2})^{\frac{5}{4}} + C

Solution steps

\int \frac{5 x ( 2 - x ^{2} ) ^{\frac{1}{4}}}{4} d x

Take the constant out:

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

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

Apply u-substitution

: \int - \frac{u ^{\frac{1}{4}}}{2} d u

= \frac{5}{4} \cdot \int - \frac{u ^{\frac{1}{4}}}{2} d u

Take the constant out:

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

= \frac{5}{4} (- \frac{1}{2} \cdot \int u^{\frac{1}{4}} d u)

Apply the Power Rule

: \frac{4}{5} u^{\frac{5}{4}}

= \frac{5}{4} (- \frac{1}{2} \cdot \frac{4}{5} u^{\frac{5}{4}})

Substitute back

u = 2 - x^{2}

= \frac{5}{4} (- \frac{1}{2} \cdot \frac{4}{5} (2 - x^{2})^{\frac{5}{4}})

Simplify

\frac{5}{4} (- \frac{1}{2} \cdot \frac{4}{5} (2 - x^{2})^{\frac{5}{4}}) : - \frac{1}{2} (2 - x^{2})^{\frac{5}{4}}

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

Add a constant to the solution

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

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What is the integral of (5x(2-x^2)^{1/4})/4 ?
The integral of (5x(2-x^2)^{1/4})/4 is -1/2 (2-x^2)^{5/4}+C