What is the integral of a/(bx+c) ?

The integral of a/(bx+c) is a 1/b ln|bx+c|+C

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integral of a/(bx+c)

\int \frac{a}{b x + c} d x

a \frac{1}{b} ln ∣ b x + c ∣ + C

Solution steps

\int \frac{a}{b x + c} d x

Take the constant out:

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

= a \cdot \int \frac{1}{b x + c} d x

Apply u-substitution

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

= a \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

= a \frac{1}{b} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= a \frac{1}{b} ln ∣ u ∣

Substitute back

u = b x + c

= a \frac{1}{b} ln ∣ b x + c ∣

Add a constant to the solution

= a \frac{1}{b} ln ∣ b x + c ∣ + C

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