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

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

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

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

- \frac{1}{2 x ^{2}} + C

Solution steps

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

Apply exponent rule:

\frac{1}{a ^{b}} = a^{- b}

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

= \int x^{- 3} \frac{1}{( 1 ) ^{\frac{1}{3}}} d x

Simplify

x^{- 3} \frac{1}{( 1 ) ^{\frac{1}{3}}} : \frac{1}{x ^{3}}

= \int \frac{1}{x ^{3}} d x

Apply the Power Rule

: - \frac{1}{2 x ^{2}}

= - \frac{1}{2 x ^{2}}

Add a constant to the solution

= - \frac{1}{2 x ^{2}} + C

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