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

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

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

Solution steps

Apply u-substitution

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

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

Apply u-substitution

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

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

Expand

\frac{( v + 1 ) ^{2}}{v} : v + 2 + \frac{1}{v}

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

\int v d v = \frac{v ^{2}}{2}

\int 2 d v = 2 v

\int \frac{1}{v} d v = ln ∣ v ∣

= 3 (\frac{v ^{2}}{2} + 2 v + ln ∣ v ∣)

Substitute back

Add a constant to the solution

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

\int \frac{tan ( x )}{( cos ( x ) ) ^{2}} d x

\int (8 x - 3) d x

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

\int x^{6} + \frac{1}{x ^{6}} d x

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