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

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

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

\int \frac{x ^{- \frac{1}{3}}}{1 + x ^{\frac{2}{3}}} d x

\frac{3}{2} ln 1 + x^{\frac{2}{3}} + C

Solution steps

\int \frac{x ^{- \frac{1}{3}}}{1 + x ^{\frac{2}{3}}} d x

Apply u-substitution

: \int \frac{3}{2 u} d u

= \int \frac{3}{2 u} d u

Take the constant out:

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

= \frac{3}{2} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{3}{2} ln ∣ u ∣

Substitute back

u = 1 + x^{\frac{2}{3}}

= \frac{3}{2} ln 1 + x^{\frac{2}{3}}

Add a constant to the solution

= \frac{3}{2} ln 1 + x^{\frac{2}{3}} + C

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