What is the integral of (x+3)/(sqrt(x^2+4x+5)) ?

The integral of (x+3)/(sqrt(x^2+4x+5)) is sqrt(x^2+4x+5)+ln|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

Popular >

integral of (x+3)/(sqrt(x^2+4x+5))

\int \frac{x + 3}{x ^{2} + 4 x + 5} d x

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

Solution steps

\int \frac{x + 3}{x ^{2} + 4 x + 5} d x

Expand

\frac{x + 3}{x ^{2} + 4 x + 5} : \frac{x}{x ^{2} + 4 x + 5} + \frac{3}{x ^{2} + 4 x + 5}

= \int \frac{x}{x ^{2} + 4 x + 5} + \frac{3}{x ^{2} + 4 x + 5} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{x}{x ^{2} + 4 x + 5} d x + \int \frac{3}{x ^{2} + 4 x + 5} d x

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

\int \frac{3}{x ^{2} + 4 x + 5} d x = 3 ln x^{2} + 4 x + 5 + x + 2

= x^{2} + 4 x + 5 - 2 ln x^{2} + 4 x + 5 + x + 2 + 3 ln x^{2} + 4 x + 5 + x + 2

Simplify

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

Add a constant to the solution

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

ln (7 x^{2} + 3 y^{2})

\int cot^{3} (3 x) csc^{5} (3 x) d x

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

\int (1 - y^{2})^{2} d y

y = \frac{1}{( 7 x + 12 ) ^{3}}

What is the integral of (x+3)/(sqrt(x^2+4x+5)) ?
The integral of (x+3)/(sqrt(x^2+4x+5)) is sqrt(x^2+4x+5)+ln|sqrt(x^2+4x+5)+x+2|+C