What is the integral of 1/(2+sqrt(x)) ?

The integral of 1/(2+sqrt(x)) is 2(2+sqrt(x)-2ln|2+sqrt(x)|)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 1/(2+sqrt(x))

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

2 (2 + x - 2 ln 2 + x) + C

Solution steps

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

Apply u-substitution

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

= \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} : 1 - \frac{2}{u}

= 2 \cdot \int 1 - \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 1 d u - \int \frac{2}{u} d u)

\int 1 d u = u

\int \frac{2}{u} d u = 2 ln ∣ u ∣

= 2 (u - 2 ln ∣ u ∣)

Substitute back

u = 2 + x

= 2 (2 + x - 2 ln 2 + x)

Add a constant to the solution

= 2 (2 + x - 2 ln 2 + x) + C

y = cot (7 x - 3)

x \to 4 lim (\frac{6 x}{x - 4})

\int (\frac{6}{x} + 6 x) d x

\int_{6}^{10} ln (x^{13}) d x

\int (4 x^{3}) d x

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