What is the integral of (\sqrt[4]{x})/(1+sqrt(x)) ?

The integral of (\sqrt[4]{x})/(1+sqrt(x)) is 4(1/3 x^{3/4}-\sqrt[4]{x}+arctan(\sqrt[4]{x}))+C

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integral of (\sqrt[4]{x})/(1+sqrt(x))

Solution steps

Apply u-substitution

: \int \frac{4 u ^{4}}{1 + u ^{2}} d u

= \int \frac{4 u ^{4}}{1 + u ^{2}} d u

Take the constant out:

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

= 4 \cdot \int \frac{u ^{4}}{1 + u ^{2}} d u

Long division

\frac{u ^{4}}{1 + u ^{2}} : u^{2} - 1 + \frac{1}{u ^{2} + 1}

= 4 \cdot \int u^{2} - 1 + \frac{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

= 4 (\int u^{2} d u - \int 1 d u + \int \frac{1}{u ^{2} + 1} d u)

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

\int 1 d u = u

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

= 4 (\frac{u ^{3}}{3} - u + arctan (u))

Substitute back

Simplify

Add a constant to the solution

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

\int \frac{x ^{2} + 33}{x ^{2} + 36} d x

\int \frac{2 x ^{2} + 5 x - 1}{x ^{3} + x ^{2} - 2 x} d x

\int \frac{1}{( 3 + sin ^{2} ( x ) )} d x

\int (x 4 - x^{2}) d x

What is the integral of (\sqrt[4]{x})/(1+sqrt(x)) ?
The integral of (\sqrt[4]{x})/(1+sqrt(x)) is 4(1/3 x^{3/4}-\sqrt[4]{x}+arctan(\sqrt[4]{x}))+C