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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

integral

\frac{1}{( x + 2 )}

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

Solution steps

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

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

Add a constant to the solution

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

cos^{- 2} (x)

1 - cos (6 x)

(5 x - 4)^{- 3}

∣ x ∣^{2}

sin (y^{3})

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