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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int - \frac{1}{6 u} d u

= 2 \cdot \int - \frac{1}{6 u} d u

Take the constant out:

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

= 2 (- \frac{1}{6} \cdot \int \frac{1}{u} d u)

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 (- \frac{1}{6} ln ∣ u ∣)

Substitute back

u = 2 - 3 v^{2}

= 2 (- \frac{1}{6} ln 2 - 3 v^{2})

Simplify

2 (- \frac{1}{6} ln 2 - 3 v^{2}) : - \frac{1}{3} ln 2 - 3 v^{2}

= - \frac{1}{3} ln 2 - 3 v^{2}

Add a constant to the solution

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

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