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

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

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

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

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

Solution steps

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

Apply u-substitution

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

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

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

Add a constant to the solution

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

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