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

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

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

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

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

Solution steps

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

Apply u-substitution

= \int \frac{3 u}{1 + u} d u

Take the constant out:

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

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

Apply u-substitution

= 3 \cdot \int \frac{v - 1}{v} d v

Expand

\frac{v - 1}{v} : 1 - \frac{1}{v}

= 3 \cdot \int 1 - \frac{1}{v} d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int 1 d v = v

\int \frac{1}{v} d v = ln ∣ v ∣

= 3 (v - ln ∣ v ∣)

Substitute back

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

Add a constant to the solution

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

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