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

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

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

Apply u-substitution

= 2 \cdot \int 2 d v

Integral of a constant:

\int a d x = a x

= 2 \cdot 2 v

Substitute back

= 2 \cdot 2 1 + x

Simplify

= 4 1 + x

Add a constant to the solution

= 4 1 + x + C

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