What is the integral of 3/(2-5x) ?

The integral of 3/(2-5x) is -3/5 ln|2-5x|+C

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integral of 3/(2-5x)

\int \frac{3}{2 - 5 x} d x

- \frac{3}{5} ln ∣ 2 - 5 x ∣ + C

Solution steps

\int \frac{3}{2 - 5 x} d x

Take the constant out:

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

= 3 \cdot \int \frac{1}{2 - 5 x} d x

Apply u-substitution

= 3 \cdot \int - \frac{1}{5 u} d u

Take the constant out:

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

= 3 (- \frac{1}{5} \cdot \int \frac{1}{u} d u)

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 3 (- \frac{1}{5} ln ∣ u ∣)

Substitute back

u = 2 - 5 x

= 3 (- \frac{1}{5} ln ∣ 2 - 5 x ∣)

Simplify

3 (- \frac{1}{5} ln ∣ 2 - 5 x ∣) : - \frac{3}{5} ln ∣ 2 - 5 x ∣

= - \frac{3}{5} ln ∣ 2 - 5 x ∣

Add a constant to the solution

= - \frac{3}{5} ln ∣ 2 - 5 x ∣ + C