What is the integral of 1/(1-0.4v) ?

The integral of 1/(1-0.4v) is -2.5ln|1-0.4v|+C

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integral of 1/(1-0.4v)

\int \frac{1}{1 - 0.4 v} d v

- 2.5 ln ∣ 1 - 0.4 v ∣ + C

Solution steps

\int \frac{1}{1 - 0.4 v} d v

Apply u-substitution

= \int - \frac{1}{0.4 u} d u

Take the constant out:

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

= - \frac{1}{0.4} \cdot \int \frac{1}{u} d u

Simplify

= - 2.5 \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - 2.5 ln ∣ u ∣

Substitute back

u = 1 - 0.4 v

= - 2.5 ln ∣ 1 - 0.4 v ∣

Add a constant to the solution

= - 2.5 ln ∣ 1 - 0.4 v ∣ + C

What is the integral of 1/(1-0.4v) ?
The integral of 1/(1-0.4v) is -2.5ln|1-0.4v|+C