What is the integral of (2v+6)/(7v-12) ?

The integral of (2v+6)/(7v-12) is 2/7 v-24/49+66/49 ln|7v-12|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (2v+6)/(7v-12)

\int \frac{2 v + 6}{7 v - 12} d v

\frac{2}{7} v - \frac{24}{49} + \frac{66}{49} ln ∣ 7 v - 12 ∣ + C

Solution steps

\int \frac{2 v + 6}{7 v - 12} d v

Expand

\frac{2 v + 6}{7 v - 12} : \frac{2 v}{7 v - 12} + \frac{6}{7 v - 12}

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{2 v}{7 v - 12} d v + \int \frac{6}{7 v - 12} d v

\int \frac{2 v}{7 v - 12} d v = \frac{2}{49} (7 v - 12 + 12 ln ∣ 7 v - 12 ∣)

\int \frac{6}{7 v - 12} d v = \frac{6}{7} ln ∣ 7 v - 12 ∣

= \frac{2}{49} (7 v - 12 + 12 ln ∣ 7 v - 12 ∣) + \frac{6}{7} ln ∣ 7 v - 12 ∣

Simplify

\frac{2}{49} (7 v - 12 + 12 ln ∣ 7 v - 12 ∣) + \frac{6}{7} ln ∣ 7 v - 12 ∣ : \frac{2}{7} v - \frac{24}{49} + \frac{66}{49} ln ∣ 7 v - 12 ∣

= \frac{2}{7} v - \frac{24}{49} + \frac{66}{49} ln ∣ 7 v - 12 ∣

Add a constant to the solution

= \frac{2}{7} v - \frac{24}{49} + \frac{66}{49} ln ∣ 7 v - 12 ∣ + C

\int \frac{8 - 3 x e ^{x}}{x} d x

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

\int (x^{3} - e^{x^{4}}) d x

\int \frac{( 3 x ^{2} + 9 x + 1 )}{x} d x

\int \frac{sin ( 2 x )}{2 + 3 cos ( 2 x )} d x

What is the integral of (2v+6)/(7v-12) ?
The integral of (2v+6)/(7v-12) is 2/7 v-24/49+66/49 ln|7v-12|+C