What is the integral of (3+v)/(1-v^2) ?

The integral of (3+v)/(1-v^2) is 3((ln|v+1|)/2-(ln|v-1|)/2)-1/2 ln|1-v^2|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (3+v)/(1-v^2)

\int \frac{3 + v}{1 - v ^{2}} d v

3 (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}) - \frac{1}{2} ln 1 - v^{2} + C

Solution steps

\int \frac{3 + v}{1 - v ^{2}} d v

Expand

\frac{3 + v}{1 - v ^{2}} : \frac{3}{1 - v ^{2}} + \frac{v}{1 - v ^{2}}

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{3}{1 - v ^{2}} d v + \int \frac{v}{1 - v ^{2}} d v

\int \frac{3}{1 - v ^{2}} d v = 3 (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2})

\int \frac{v}{1 - v ^{2}} d v = - \frac{1}{2} ln 1 - v^{2}

= 3 (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}) - \frac{1}{2} ln 1 - v^{2}

Add a constant to the solution

= 3 (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}) - \frac{1}{2} ln 1 - v^{2} + C

What is the integral of (3+v)/(1-v^2) ?
The integral of (3+v)/(1-v^2) is 3((ln|v+1|)/2-(ln|v-1|)/2)-1/2 ln|1-v^2|+C