What is the integral of (x^4)/((1+x^5)^{2/3)} ?

The integral of (x^4)/((1+x^5)^{2/3)} is 3/5 (1+x^5)^{1/3}+C

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integral of (x^4)/((1+x^5)^{2/3)}

\int \frac{x ^{4}}{( 1 + x ^{5} ) ^{\frac{2}{3}}} d x

\frac{3}{5} (1 + x^{5})^{\frac{1}{3}} + C

Solution steps

\int \frac{x ^{4}}{( 1 + x ^{5} ) ^{\frac{2}{3}}} d x

Apply u-substitution

: \int \frac{1}{5 u ^{\frac{2}{3}}} d u

= \int \frac{1}{5 u ^{\frac{2}{3}}} 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 \frac{1}{u ^{\frac{2}{3}}} d u

Apply the Power Rule

: 3 u^{\frac{1}{3}}

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

Substitute back

u = 1 + x^{5}

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

Simplify

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

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

Add a constant to the solution

= \frac{3}{5} (1 + x^{5})^{\frac{1}{3}} + C

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What is the integral of (x^4)/((1+x^5)^{2/3)} ?
The integral of (x^4)/((1+x^5)^{2/3)} is 3/5 (1+x^5)^{1/3}+C