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

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

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

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

4 (\frac{1}{3} (1 + x)^{\frac{3}{2}} - 1 + x) + C

Solution steps

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

Apply u-substitution

: \int 4 (u^{2} - 1) d u

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

Take the constant out:

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

= 4 \cdot \int 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 u^{2} d u = \frac{u ^{3}}{3}

\int 1 d u = u

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

Substitute back

u = 1 + x

= 4 \frac{( 1 + x ) ^{3}}{3} - 1 + x

Simplify

4 \frac{( 1 + x ) ^{3}}{3} - 1 + x : 4 (\frac{1}{3} (1 + x)^{\frac{3}{2}} - 1 + x)

= 4 (\frac{1}{3} (1 + x)^{\frac{3}{2}} - 1 + x)

Add a constant to the solution

= 4 (\frac{1}{3} (1 + x)^{\frac{3}{2}} - 1 + x) + C

(0, - 6), (3, - 5)

t \to 0 lim (\frac{sin ( 3 t ) + 4 t}{t sec ( t )})

\frac{\partial}{\partial y} (5 x ln (x y))

\frac{\partial}{\partial x} (y^{2} cos (x))

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