Integrate ∫ x/((x - 2)(x - 4)) dx Step by Step

Step by step

The question

Question: find the integral of x, all over (x − 2)(x − 4).

∫ x(x − 2)(x − 4) dx =

1 Split the fraction

Two simple fractions can add up to this one: A over (x − 2), plus B over (x − 4).

We find the numbers A and B.

x(x − 2)(x − 4) = A(x − 2) + B(x − 4)

Careful!

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

Here it is not, so we split.

Careful: not ∫ x(x − 2)(x − 4) dx = ln|(x − 2)(x − 4)| ✗

2 Clear the bottoms

Multiply both sides by (x − 2)(x − 4).

On the left the bottom goes.

On the right each fraction keeps the other bracket.

x = A(x − 4) + B(x − 2)

3 Find A and B

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

x = 2: 2 = A(−2) + B(0)

Divide by −2: A = −1.

A = 2−2 = −1

Put in x = 4: it makes (x − 4) zero, so the A part is gone.

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

Divide by 2: B = 2.

B = 42 = 2

4 Integrate

Now the integral is two easy pieces.

∫ x(x − 2)(x − 4) dx = ∫ (−1(x − 2) + 2(x − 4)) 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 − 2| + 2 ln|x − 4| + C.

= −ln|x − 2| + 2ln|x − 4| + C

5 Check

Let's check: add the two fractions back over one bottom, (x − 2)(x − 4).

−1(x − 2) + 2(x − 4) = −(x − 4) + 2(x − 2)(x − 2)(x − 4)

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

Right!

= x(x − 2)(x − 4)

Let's look back.

First: split the fraction.

Then: clear the bottoms.

Then: find A and B.

Then: integrate.

Last: check.

Answer