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

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

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

2 (2 + x - 1 - 2 ln 2 + x - 1) + C

Solution steps

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

Apply u-substitution

: \int \frac{1}{2 + u} d u

= \int \frac{1}{2 + u} d u

Apply u-substitution

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

= \int \frac{2 ( v - 2 )}{v} d v

Take the constant out:

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

= 2 \cdot \int \frac{v - 2}{v} d v

Expand

\frac{v - 2}{v} : 1 - \frac{2}{v}

= 2 \cdot \int 1 - \frac{2}{v} d v

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 v - \int \frac{2}{v} d v)

\int 1 d v = v

\int \frac{2}{v} d v = 2 ln ∣ v ∣

= 2 (v - 2 ln ∣ v ∣)

Substitute back

= 2 (2 + x - 1 - 2 ln 2 + x - 1)

Add a constant to the solution

= 2 (2 + x - 1 - 2 ln 2 + x - 1) + C

\int x (x + 3) d x

\int \frac{3 - 2 x}{1 - x ^{2}} d x

\int (4 x^{5} + 5 x^{3} + 11) d x

\int cos^{3} (\frac{1}{2} x) d x

\int \frac{8 x}{x ^{2} - 4} d x

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