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

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

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

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

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

Solution steps

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

Expand

\frac{2 - v}{v ^{2} - 1} : \frac{2}{v ^{2} - 1} - \frac{v}{v ^{2} - 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{2}{v ^{2} - 1} d v - \int \frac{v}{v ^{2} - 1} d v

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

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

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

Add a constant to the solution

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

f (x) = x - 2, g (x) = - x^{2} + x + 2

\int x^{2} x - 4 d x

\int \frac{sin ^{3} ( x )}{cos ^{5} ( x )} d x

\frac{\partial}{\partial y} (e^{3 x - y^{2}})

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