What is the integral of 3/(a-bx) ?

The integral of 3/(a-bx) is -3/b ln|a-bx|+C

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integral of 3/(a-bx)

\int \frac{3}{a - b x} d x

- \frac{3}{b} ln ∣ a - b x ∣ + C

Solution steps

\int \frac{3}{a - b 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}{a - b x} d x

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 3 (- \frac{1}{b} ln ∣ u ∣)

Substitute back

u = a - b x

= 3 (- \frac{1}{b} ln ∣ a - b x ∣)

Simplify

3 (- \frac{1}{b} ln ∣ a - b x ∣) : - \frac{3}{b} ln ∣ a - b x ∣

= - \frac{3}{b} ln ∣ a - b x ∣

Add a constant to the solution

= - \frac{3}{b} ln ∣ a - b x ∣ + C

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