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

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

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integral of 1/(1-3v)

\int \frac{1}{1 - 3 v} d v

- \frac{1}{3} ln ∣ 1 - 3 v ∣ + C

Solution steps

\int \frac{1}{1 - 3 v} d v

Apply u-substitution

: \int - \frac{1}{3 u} d u

= \int - \frac{1}{3 u} d u

Take the constant out:

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

= - \frac{1}{3} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - \frac{1}{3} ln ∣ u ∣

Substitute back

u = 1 - 3 v

= - \frac{1}{3} ln ∣ 1 - 3 v ∣

Add a constant to the solution

= - \frac{1}{3} ln ∣ 1 - 3 v ∣ + C

x \to 9 lim (\frac{x ^{2}}{3} - \frac{3}{x})

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