What is the integral of 2/(sqrt(x^2+4x+5)) ?

The integral of 2/(sqrt(x^2+4x+5)) is 2ln|sqrt(x^2+4x+5)+x+2|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of 2/(sqrt(x^2+4x+5))

\int \frac{2}{x ^{2} + 4 x + 5} d x

2 ln x^{2} + 4 x + 5 + x + 2 + C

Solution steps

\int \frac{2}{x ^{2} + 4 x + 5} d x

Take the constant out:

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

= 2 \cdot \int \frac{1}{x ^{2} + 4 x + 5} d x

Complete the square

x^{2} + 4 x + 5 : (x + 2)^{2} + 1

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

Apply u-substitution

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

Use the common integral:

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

= 2 ln u^{2} + 1 + u

Substitute back

u = x + 2

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

Expand

(x + 2)^{2} + 1 : x^{2} + 4 x + 5

= 2 ln x^{2} + 4 x + 5 + x + 2

Add a constant to the solution

= 2 ln x^{2} + 4 x + 5 + x + 2 + C

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

\int - 6 x^{6} - 4 x^{5} + 5 x - 3 d x

\int (16 x + 5) d x

\int (ln (7 x)) d x

\int 4 x + 10 d x

What is the integral of 2/(sqrt(x^2+4x+5)) ?
The integral of 2/(sqrt(x^2+4x+5)) is 2ln|sqrt(x^2+4x+5)+x+2|+C