What is the integral of 1/(sqrt(2+\sqrt{7+x))} ?

The integral of 1/(sqrt(2+\sqrt{7+x))} is 2(2/3 (2+sqrt(7+x))^{3/2}-4sqrt(2+\sqrt{7+x)})+C

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integral of 1/(sqrt(2+\sqrt{7+x))}

\int \frac{1}{2 + 7 + x} d x

2 (\frac{2}{3} (2 + 7 + x)^{\frac{3}{2}} - 4 2 + 7 + x) + C

Solution steps

\int \frac{1}{2 + 7 + x} d x

Apply u-substitution

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

Expand

\frac{u - 2}{u} : u - \frac{2}{u}

= 2 \cdot \int u - \frac{2}{u} 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

= 2 (\int u d u - \int \frac{2}{u} d u)

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

\int \frac{2}{u} d u = 4 u

= 2 (\frac{2}{3} u^{\frac{3}{2}} - 4 u)

Substitute back

u = 2 + 7 + x

= 2 (\frac{2}{3} (2 + 7 + x)^{\frac{3}{2}} - 4 2 + 7 + x)

Add a constant to the solution

= 2 (\frac{2}{3} (2 + 7 + x)^{\frac{3}{2}} - 4 2 + 7 + x) + C

What is the integral of 1/(sqrt(2+\sqrt{7+x))} ?
The integral of 1/(sqrt(2+\sqrt{7+x))} is 2(2/3 (2+sqrt(7+x))^{3/2}-4sqrt(2+\sqrt{7+x)})+C