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

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

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

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

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

\int \frac{1}{sin ^{3} ( x ) cos ( x )} d x

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