What is the integral of x^4*\sqrt[4]{x^5-7} ?

The integral of x^4*\sqrt[4]{x^5-7} is 4/25 (x^5-7)^{5/4}+C

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integral of x^4*\sqrt[4]{x^5-7}

\int x^{4} \cdot 4 x^{5} - 7 d x

\frac{4}{25} (x^{5} - 7)^{\frac{5}{4}} + C

Solution steps

\int x^{4} 4 x^{5} - 7 d x

Apply u-substitution

: \int \frac{4 u}{5} d u

= \int \frac{4 u}{5} d u

Take the constant out:

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

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

Apply the Power Rule

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

= \frac{1}{5} \cdot \frac{4}{5} u^{\frac{5}{4}}

Substitute back

u = x^{5} - 7

= \frac{1}{5} \cdot \frac{4}{5} (x^{5} - 7)^{\frac{5}{4}}

Simplify

\frac{1}{5} \cdot \frac{4}{5} (x^{5} - 7)^{\frac{5}{4}} : \frac{4}{25} (x^{5} - 7)^{\frac{5}{4}}

= \frac{4}{25} (x^{5} - 7)^{\frac{5}{4}}

Add a constant to the solution

= \frac{4}{25} (x^{5} - 7)^{\frac{5}{4}} + C

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What is the integral of x^4*\sqrt[4]{x^5-7} ?
The integral of x^4*\sqrt[4]{x^5-7} is 4/25 (x^5-7)^{5/4}+C