What is the integral of (ax)/(x^2-px) ?

The integral of (ax)/(x^2-px) is aln|x-p|+C

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integral of (ax)/(x^2-px)

\int \frac{a x}{x ^{2} - p x} d x

a ln ∣ x - p ∣ + C

Solution steps

\int \frac{a x}{x ^{2} - p x} d x

Take the constant out:

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

= a \cdot \int \frac{x}{x ^{2} - p x} d x

Simplify

\frac{x}{x ^{2} - p x} : \frac{1}{x - p}

= a \cdot \int \frac{1}{x - p} d x

Apply u-substitution

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= a ln ∣ u ∣

Substitute back

u = x - p

= a ln ∣ x - p ∣

Add a constant to the solution

= a ln ∣ x - p ∣ + C

What is the integral of (ax)/(x^2-px) ?
The integral of (ax)/(x^2-px) is aln|x-p|+C