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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int \frac{3}{2 u ^{\frac{1}{2}} ( u + 1 )} d u

= \int \frac{3}{2 u ^{\frac{1}{2}} ( u + 1 )} 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 ^{\frac{1}{2}} ( u + 1 )} d u

Apply u-substitution

: \int \frac{2}{v ^{2} + 1} d v

= \frac{3}{2} \cdot \int \frac{2}{v ^{2} + 1} d v

Take the constant out:

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

= \frac{3}{2} \cdot 2 \cdot \int \frac{1}{v ^{2} + 1} d v

Use the common integral:

\int \frac{1}{v ^{2} + 1} d v = arctan (v)

= \frac{3}{2} \cdot 2 arctan (v)

Substitute back

= \frac{3}{2} \cdot 2 arctan ((x^{\frac{2}{3}})^{\frac{1}{2}})

Simplify

\frac{3}{2} \cdot 2 arctan ((x^{\frac{2}{3}})^{\frac{1}{2}}) : 3 arctan (x^{\frac{1}{3}})

= 3 arctan (x^{\frac{1}{3}})

Add a constant to the solution

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

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