What is the integral of 1/(x+\sqrt[3]{x)} ?

The integral of 1/(x+\sqrt[3]{x)} is 3/2 ln|x^{2/3}+1|+C

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integral of 1/(x+\sqrt[3]{x)}

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

Solution steps

Apply u-substitution

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

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

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 3 \cdot \frac{1}{2} ln ∣ v ∣

Substitute back

Simplify

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

Add a constant to the solution

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

6 x^{4} - 3

\int x^{4} (x^{5} - 4)^{8} d x

f (x) = \frac{3}{3 x + 1}

\frac{d}{d x} (\frac{x ^{5} + 2 x + 6}{x ^{\frac{1}{3}}})

f (x) = cos (5 x^{2})

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