What is the integral of ((x+1))/(7x+x^2) ?

The integral of ((x+1))/(7x+x^2) is 1/7 ln|x|+6/7 ln|x+7|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of ((x+1))/(7x+x^2)

\int \frac{( x + 1 )}{7 x + x ^{2}} d x

\frac{1}{7} ln ∣ x ∣ + \frac{6}{7} ln ∣ x + 7 ∣ + C

Solution steps

\int \frac{x + 1}{7 x + x ^{2}} d x

Take the partial fraction of

\frac{x + 1}{7 x + x ^{2}} : \frac{1}{7 x} + \frac{6}{7 ( x + 7 )}

= \int \frac{1}{7 x} + \frac{6}{7 ( x + 7 )} 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{1}{7 x} d x + \int \frac{6}{7 ( x + 7 )} d x

\int \frac{1}{7 x} d x = \frac{1}{7} ln ∣ x ∣

\int \frac{6}{7 ( x + 7 )} d x = \frac{6}{7} ln ∣ x + 7 ∣

= \frac{1}{7} ln ∣ x ∣ + \frac{6}{7} ln ∣ x + 7 ∣

Add a constant to the solution

= \frac{1}{7} ln ∣ x ∣ + \frac{6}{7} ln ∣ x + 7 ∣ + C

What is the integral of ((x+1))/(7x+x^2) ?
The integral of ((x+1))/(7x+x^2) is 1/7 ln|x|+6/7 ln|x+7|+C