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

The integral of 1/(sqrt(x)(1+\sqrt[4]{x))} is 4(1+\sqrt[4]{x}-ln|1+\sqrt[4]{x}|)+C

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

Solution steps

Expand

= \int \frac{1}{x + x ^{\frac{3}{4}}} d x

Apply u-substitution

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

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

Apply u-substitution

: \int \frac{v - 1}{v} d v

= 4 \cdot \int \frac{v - 1}{v} d v

Expand

\frac{v - 1}{v} : 1 - \frac{1}{v}

= 4 \cdot \int 1 - \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

= 4 (\int 1 d v - \int \frac{1}{v} d v)

\int 1 d v = v

\int \frac{1}{v} d v = ln ∣ v ∣

= 4 (v - ln ∣ v ∣)

Substitute back

Add a constant to the solution

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