What is the integral of (8x^3)/(sqrt(5-x^4)) ?

The integral of (8x^3)/(sqrt(5-x^4)) is -4sqrt(5-x^4)+C

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integral of (8x^3)/(sqrt(5-x^4))

\int \frac{8 x ^{3}}{5 - x ^{4}} d x

- 4 5 - x^{4} + C

Solution steps

\int \frac{8 x ^{3}}{5 - x ^{4}} d x

Take the constant out:

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

= 8 \cdot \int \frac{x ^{3}}{5 - x ^{4}} d x

Apply u-substitution

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

= 8 \cdot \int - \frac{1}{4 u} d u

Take the constant out:

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

= 8 (- \frac{1}{4} \cdot \int \frac{1}{u} d u)

Apply the Power Rule

: 2 u^{\frac{1}{2}}

= 8 (- \frac{1}{4} \cdot 2 u^{\frac{1}{2}})

Substitute back

u = 5 - x^{4}

= 8 (- \frac{1}{4} \cdot 2 (5 - x^{4})^{\frac{1}{2}})

Simplify

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

= - 4 5 - x^{4}

Add a constant to the solution

= - 4 5 - x^{4} + C

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What is the integral of (8x^3)/(sqrt(5-x^4)) ?
The integral of (8x^3)/(sqrt(5-x^4)) is -4sqrt(5-x^4)+C