What is the integral of (8x^2)/((5x^3+2)^{3/5)} ?

The integral of (8x^2)/((5x^3+2)^{3/5)} is 4/3 (5x^3+2)^{2/5}+C

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integral of (8x^2)/((5x^3+2)^{3/5)}

\int \frac{8 x ^{2}}{( 5 x ^{3} + 2 ) ^{\frac{3}{5}}} d x

\frac{4}{3} (5 x^{3} + 2)^{\frac{2}{5}} + C

Solution steps

\int \frac{8 x ^{2}}{( 5 x ^{3} + 2 ) ^{\frac{3}{5}}} 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 ^{2}}{( 5 x ^{3} + 2 ) ^{\frac{3}{5}}} d x

Apply u-substitution

: \int \frac{1}{15 u ^{\frac{3}{5}}} d u

= 8 \cdot \int \frac{1}{15 u ^{\frac{3}{5}}} d u

Take the constant out:

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

= 8 \cdot \frac{1}{15} \cdot \int \frac{1}{u ^{\frac{3}{5}}} d u

Apply the Power Rule

: \frac{5}{2} u^{\frac{2}{5}}

= 8 \cdot \frac{1}{15} \cdot \frac{5}{2} u^{\frac{2}{5}}

Substitute back

u = 5 x^{3} + 2

= 8 \cdot \frac{1}{15} \cdot \frac{5}{2} (5 x^{3} + 2)^{\frac{2}{5}}

Simplify

8 \cdot \frac{1}{15} \cdot \frac{5}{2} (5 x^{3} + 2)^{\frac{2}{5}} : \frac{4}{3} (5 x^{3} + 2)^{\frac{2}{5}}

= \frac{4}{3} (5 x^{3} + 2)^{\frac{2}{5}}

Add a constant to the solution

= \frac{4}{3} (5 x^{3} + 2)^{\frac{2}{5}} + C

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What is the integral of (8x^2)/((5x^3+2)^{3/5)} ?
The integral of (8x^2)/((5x^3+2)^{3/5)} is 4/3 (5x^3+2)^{2/5}+C