What is the integral of v/(0.122-0.0007v^2) ?

The integral of v/(0.122-0.0007v^2) is -714.28571…ln|0.122-0.0007v^2|+C

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integral of v/(0.122-0.0007v^2)

\int \frac{v}{0.122 - 0.0007 v ^{2}} d v

- 714.28571 \dots ln 0.122 - 0.0007 v^{2} + C

Solution steps

\int \frac{v}{0.122 - 0.0007 v ^{2}} d v

Apply u-substitution

: \int - \frac{1}{0.0014 u} d u

= \int - \frac{1}{0.0014 u} d u

Take the constant out:

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

= - \frac{1}{0.0014} \cdot \int \frac{1}{u} d u

Simplify

= - 714.28571 \dots \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - 714.28571 \dots ln ∣ u ∣

Substitute back

u = 0.122 - 0.0007 v^{2}

= - 714.28571 \dots ln 0.122 - 0.0007 v^{2}

Add a constant to the solution

= - 714.28571 \dots ln 0.122 - 0.0007 v^{2} + C

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What is the integral of v/(0.122-0.0007v^2) ?
The integral of v/(0.122-0.0007v^2) is -714.28571…ln|0.122-0.0007v^2|+C