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

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

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

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

2 ln 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

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

\int_{0}^{\frac{π}{2}} - 9 cos^{5} (x) d x

\int \frac{2 x - 5}{x ^{2} - 5 x - 1} d x

\frac{\partial}{\partial x} (sin (x y) + x cos (y))

\frac{d}{d x} (\frac{x - x}{x ^{\frac{1}{9}}})

y = 2 x + 5, y = 14 - x^{2}, (- 1, 2)

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