What is the integral of x/((x+a)(x-a)) ?

The integral of x/((x+a)(x-a)) is 1/2 ln(|x+a||x-a|)+C

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integral of x/((x+a)(x-a))

\int \frac{x}{( x + a ) ( x - a )} d x

\frac{1}{2} ln (∣ x + a ∣ ∣ x - a ∣) + C

Solution steps

\int \frac{x}{( x + a ) ( x - a )} d x

Apply u-substitution

= \int \frac{1}{2 u} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{2} ln ∣ u ∣

Substitute back

u = (x + a) (x - a)

= \frac{1}{2} ln ∣ (x + a) (x - a) ∣

Simplify

\frac{1}{2} ln ∣ (x + a) (x - a) ∣ : \frac{1}{2} ln (∣ x + a ∣ ∣ x - a ∣)

= \frac{1}{2} ln (∣ x + a ∣ ∣ x - a ∣)

Add a constant to the solution

= \frac{1}{2} ln (∣ x + a ∣ ∣ x - a ∣) + C

What is the integral of x/((x+a)(x-a)) ?
The integral of x/((x+a)(x-a)) is 1/2 ln(|x+a||x-a|)+C