Integrate ∫ 2/((x - 1)(x + 1)) dx Step by Step

Step by step

The question

Question: find the integral of 2, all over (x − 1)(x + 1).

∫ 2(x − 1)(x + 1) dx =

1 Split the fraction

Two simple fractions can add up to this one: A over (x − 1), plus B over (x + 1).

We find the numbers A and B.

2(x − 1)(x + 1) = A(x − 1) + B(x + 1)

Careful!

ln of the bottom works only when the top is the bottom's derivative.

Here it is not, so we split.

Careful: not ∫ 2(x − 1)(x + 1) dx = ln|(x − 1)(x + 1)| ✗

2 Clear the bottoms

Multiply both sides by (x − 1)(x + 1).

On the left the bottom goes.

On the right each fraction keeps the other bracket.

2 = A(x + 1) + B(x − 1)

3 Find A and B

Put in x = 1: it makes (x − 1) zero, so the B part is gone.

x = 1: 2 = A(2) + B(0)

Divide by 2: A = 1.

A = 22 = 1

Put in x = −1: it makes (x + 1) zero, so the A part is gone.

x = −1: 2 = B(−2) + A(0)

Divide by −2: B = −1.

B = 2−2 = −1

4 Integrate

Now the integral is two easy pieces.

∫ 2(x − 1)(x + 1) dx = ∫ (1(x − 1) − 1(x + 1)) dx

One over x minus a number integrates to ln of it, like 1 over x.

∫ 1x − p dx = ln|x − p| + C

So the answer is ln|x − 1| − ln|x + 1| + C.

= ln|x − 1| − ln|x + 1| + C

5 Check

Let's check: add the two fractions back over one bottom, (x − 1)(x + 1).

1(x − 1) − 1(x + 1) = (x + 1) − (x − 1)(x − 1)(x + 1)

Open the brackets and add: the top is 2 again.

Right!

= 2(x − 1)(x + 1)

Let's look back.

First: split the fraction.

Then: clear the bottoms.

Then: find A and B.

Then: integrate.

Last: check.

Answer