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

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

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

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

Solution steps

Apply u-substitution

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

= \int \frac{3 u}{1 - u ^{2}} 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 ^{2}} 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 (- \frac{1}{2} \cdot \int \frac{1}{v} d v)

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 3 (- \frac{1}{2} ln ∣ v ∣)

Substitute back

Simplify

= - \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

\frac{8 x ^{2} + 4 x + 4}{4}

\int (e^{x} + 5) d x

x \to 3 lim (9 - x^{2})

x \to 0 lim ((ln (x))^{2})

x \to - 1 + lim (\frac{1}{x ^{2} + 1})

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