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

The integral of 1/(sqrt(x(1+\sqrt{x)))} is 4sqrt(1+\sqrt{x)}+C

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

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

4 1 + x + C

Solution steps

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

Apply u-substitution

: \int \frac{2}{u} d u

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

Take the constant out:

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

= 2 \cdot \int \frac{1}{u} d u

Apply the Power Rule

: 2 u^{\frac{1}{2}}

= 2 \cdot 2 u^{\frac{1}{2}}

Substitute back

u = 1 + x

= 2 \cdot 2 (1 + x)^{\frac{1}{2}}

Simplify

2 \cdot 2 (1 + x)^{\frac{1}{2}} : 4 1 + x

= 4 1 + x

Add a constant to the solution

= 4 1 + x + C

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