What is the integral of ((x+3))/(x^2+6x+1) ?

The integral of ((x+3))/(x^2+6x+1) is

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of ((x+3))/(x^2+6x+1)

\int \frac{( x + 3 )}{x ^{2} + 6 x + 1} d x

\frac{2 2 + 3}{4 2} ln x + 3 + 22 + \frac{2 2 - 3}{4 2} ln x + 3 - 22 - \frac{3}{4 2} (ln \frac{x + 3}{2 2} + 1 - ln \frac{x + 3}{2 2} - 1) + C

Solution steps

\int \frac{x + 3}{x ^{2} + 6 x + 1} d x

Expand

\frac{x + 3}{x ^{2} + 6 x + 1} : \frac{x}{x ^{2} + 6 x + 1} + \frac{3}{x ^{2} + 6 x + 1}

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} + 6 x + 1} d x + \int \frac{3}{x ^{2} + 6 x + 1} d x

\int \frac{x}{x ^{2} + 6 x + 1} d x = \frac{2 2 + 3}{4 2} ln x + 3 + 22 + \frac{2 2 - 3}{4 2} ln x + 3 - 22

\int \frac{3}{x ^{2} + 6 x + 1} d x = - \frac{3}{4 2} (ln \frac{x + 3}{2 2} + 1 - ln \frac{x + 3}{2 2} - 1)

= \frac{2 2 + 3}{4 2} ln x + 3 + 22 + \frac{2 2 - 3}{4 2} ln x + 3 - 22 - \frac{3}{4 2} (ln \frac{x + 3}{2 2} + 1 - ln \frac{x + 3}{2 2} - 1)

Add a constant to the solution

= \frac{2 2 + 3}{4 2} ln x + 3 + 22 + \frac{2 2 - 3}{4 2} ln x + 3 - 22 - \frac{3}{4 2} (ln \frac{x + 3}{2 2} + 1 - ln \frac{x + 3}{2 2} - 1) + C

\int \frac{x + 1}{x ^{2} + 2 x + 49} d x

\int x^{1.9} + 7 x^{2.5} d x

\int \frac{( x ^{2} )}{x ^{4} - 1} d x

\int e^{\frac{t}{4}} d t

\int e^{- 100 x} d x

What is the integral of ((x+3))/(x^2+6x+1) ?
The integral of ((x+3))/(x^2+6x+1) is