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

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

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

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

ln v^{2} - 1 + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply u-substitution

: \int \frac{1}{2 u} d u

= 2 \cdot \int \frac{1}{2 u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \frac{1}{2} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 \cdot \frac{1}{2} ln ∣ u ∣

Substitute back

u = v^{2} - 1

= 2 \cdot \frac{1}{2} ln v^{2} - 1

Simplify

2 \cdot \frac{1}{2} ln v^{2} - 1 : ln v^{2} - 1

= ln v^{2} - 1

Add a constant to the solution

= ln v^{2} - 1 + C

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