Evaluate ∫ from 2 to 4 of 1/(x^2

Step by step

The question

Question: find the integral from 2 to 4 of 1, all over x² − 1.

∫ from 2 to 4 of 1x² − 1 dx =

1 Factor the bottom

First factor the bottom.

x² − 1 is a difference of two squares: x² minus 1 squared.

So it is (x − 1)(x + 1).

x² − 1 = (x + 1)(x − 1)

a² − b² = (a − b)(a + b), b = 1

2 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.

1(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 ∫ 1x² − 1 dx = ln|x² − 1| ✗

3 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.

1 = A(x − 1) + B(x + 1)

4 Find A and B

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

x = −1: 1 = A(−2) + B(0)

Divide by −2: A = −1/2.

A = 1−2 = −12

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

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

Divide by 2: B = 1/2.

B = 12 = 12

5 Integrate

Now the integral is two easy pieces.

∫ from 2 to 4 of 1(x + 1)(x − 1) dx = ∫ from 2 to 4 of (−12(x + 1) + 12(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

No + C: top minus bottom.

= [ −12ln|x + 1| + 12ln|x − 1| ] from 2 to 4

6 Top minus bottom

Put in 4, then take away the same with 2, for each piece.

= −12(ln 5 − ln 3) + 12(ln 3 − ln 1)

Two logs taken away make one log of a fraction.

ln a − ln b = ln ab

The two numbers in front are the same size, 1/2.

Take it out: one log minus the other is the log of their fraction.

= −12ln 53 + 12ln 3

So the integral is 1/2 times ln of 9/5, about 0.294.

= 12ln 95 ≈ 0.294

7 Check

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

−12(x + 1) + 12(x − 1) = −12(x − 1) + 12(x + 1)(x + 1)(x − 1)

Open the brackets and add: the top is 1 again.

Right!

= 1(x + 1)(x − 1)

Let's look back.

First: factor the bottom.

Then: split the fraction.

Then: clear the bottoms.

Then: find A and B.

Then: integrate.

Then: top minus bottom.

Last: check.

Answer