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

The integral of 1/(sqrt(x(1+\sqrt{x))^2)} is 2ln|1+sqrt(x)|+C

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

\int \frac{1}{x ( 1 + x ) ^{2}} d x

2 ln 1 + x + C

Solution steps

\int \frac{1}{x ( 1 + x ) ^{2}} d x

Apply exponent rule:

(a \cdot b)^{n} = a^{n} \cdot b^{n}

x (1 + x)^{2} = x (1 + x)^{2}

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

(1 + x)^{2} = ((1 + x)),

assuming

(1 + x) \geq 0

= \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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 ln ∣ u ∣

Substitute back

u = 1 + x

= 2 ln 1 + x

Add a constant to the solution

= 2 ln 1 + x + C

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