What is the integral of 5x^2\sqrt[4]{x^3+7} ?

The integral of 5x^2\sqrt[4]{x^3+7} is 4/3 (x^3+7)^{5/4}+C

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integral of 5x^2\sqrt[4]{x^3+7}

\int 5 x^{2} 4 x^{3} + 7 d x

\frac{4}{3} (x^{3} + 7)^{\frac{5}{4}} + C

Solution steps

\int 5 x^{2} 4 x^{3} + 7 d x

Take the constant out:

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

= 5 \cdot \int x^{2} 4 x^{3} + 7 d x

Apply u-substitution

: \int \frac{4 u}{3} d u

= 5 \cdot \int \frac{4 u}{3} d u

Take the constant out:

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

= 5 \cdot \frac{1}{3} \cdot \int 4 u d u

Apply the Power Rule

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

= 5 \cdot \frac{1}{3} \cdot \frac{4}{5} u^{\frac{5}{4}}

Substitute back

u = x^{3} + 7

= 5 \cdot \frac{1}{3} \cdot \frac{4}{5} (x^{3} + 7)^{\frac{5}{4}}

Simplify

5 \cdot \frac{1}{3} \cdot \frac{4}{5} (x^{3} + 7)^{\frac{5}{4}} : \frac{4}{3} (x^{3} + 7)^{\frac{5}{4}}

= \frac{4}{3} (x^{3} + 7)^{\frac{5}{4}}

Add a constant to the solution

= \frac{4}{3} (x^{3} + 7)^{\frac{5}{4}} + C

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What is the integral of 5x^2\sqrt[4]{x^3+7} ?
The integral of 5x^2\sqrt[4]{x^3+7} is 4/3 (x^3+7)^{5/4}+C