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

The integral of 1/((1+\sqrt[3]{x))sqrt(x)} is 6(-arctan(\sqrt[6]{x})+\sqrt[6]{x})+C

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

Solution steps

Apply u-substitution

: \int \frac{6 u ^{2}}{1 + u ^{2}} d u

= \int \frac{6 u ^{2}}{1 + u ^{2}} d u

Take the constant out:

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

= 6 \cdot \int \frac{u ^{2}}{1 + u ^{2}} d u

\frac{u ^{2}}{1 + u ^{2}} = - \frac{1}{1 + u ^{2}} + 1

= 6 \cdot \int - \frac{1}{1 + u ^{2}} + 1 d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 6 (- \int \frac{1}{1 + u ^{2}} d u + \int 1 d u)

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

\int 1 d u = u

= 6 (- arctan (u) + u)

Substitute back

Add a constant to the solution

\int \frac{x e ^{x}}{e ^{x} - 2} d x

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

\int e^{x} (2^{x} - 1) d x

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

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

What is the integral of 1/((1+\sqrt[3]{x))sqrt(x)} ?
The integral of 1/((1+\sqrt[3]{x))sqrt(x)} is 6(-arctan(\sqrt[6]{x})+\sqrt[6]{x})+C