Evaluate ∫ from 3 to 5 of 2/(x^2

Step by step

The question

Question: find the integral from 3 to 5 of 2, all over x² − 4.

∫ from 3 to 5 of 2x² − 4 dx =

1 Factor the bottom

First factor the bottom.

x² − 4 is a difference of two squares: x² minus 2 squared.

So it is (x − 2)(x + 2).

x² − 4 = (x + 2)(x − 2)

a² − b² = (a − b)(a + b), b = 2

2 Split the fraction

Two simple fractions can add up to this one: A over (x + 2), plus B over (x − 2).

We find the numbers A and B.

2(x + 2)(x − 2) = A(x + 2) + B(x − 2)

Careful!

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

Here it is not, so we split.

Careful: not ∫ 2x² − 4 dx = ln|x² − 4| ✗

3 Clear the bottoms

Multiply both sides by (x + 2)(x − 2).

On the left the bottom goes.

On the right each fraction keeps the other bracket.

2 = A(x − 2) + B(x + 2)

4 Find A and B

Put in x = −2: it makes (x + 2) zero, so the B part is gone.

x = −2: 2 = A(−4) + B(0)

Divide by −4: A = −1/2.

A = 2−4 = −12

Put in x = 2: it makes (x − 2) zero, so the A part is gone.

x = 2: 2 = B(4) + A(0)

Divide by 4: B = 1/2.

B = 24 = 12

5 Integrate

Now the integral is two easy pieces.

∫ from 3 to 5 of 2(x + 2)(x − 2) dx = ∫ from 3 to 5 of (−12(x + 2) + 12(x − 2)) 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 + 2| + 12ln|x − 2| ] from 3 to 5

6 Top minus bottom

Put in 5, then take away the same with 3, for each piece.

= −12(ln 7 − ln 5) + 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 75 + 12ln 3

So the integral is 1/2 times ln of 15/7, about 0.381.

= 12ln 157 ≈ 0.381

7 Check

Let's check: add the two fractions back over one bottom, (x + 2)(x − 2).

−12(x + 2) + 12(x − 2) = −12(x − 2) + 12(x + 2)(x + 2)(x − 2)

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

Right!

= 2(x + 2)(x − 2)

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