Evaluate ∫ from 1 to 3 of 2/x dx Step by Step

Step by step

The question

Question: find the integral from 1 to 3 of 2 over x.

∫ from 1 to 3 of 2x dx =

1 The rules

Now 1 over x.

The power rule cannot do it: x to the −1 would become x to the 0, over 0.

Instead: the derivative of ln x is 1 over x.

So the integral of 1/x is ln|x| + C.

∫ 1x dx = ln|x| + C

The bars mean the size of x.

ln only takes positive numbers, and x may be negative.

|x|: ln needs a number > 0

Careful!

The power rule fails for 1/x: the new power would be 0, and we cannot divide by 0.

Use ln.

Careful: not x⁰0 ✗

2 Integrate

2 times 1 over the bottom gives 2 ln|x|.

No + C: we work out top minus bottom.

∫ from 1 to 3 of 2x dx = [ 2ln|x| ] from 1 to 3

3 Top minus bottom

Put in 3, then take away the same with 1.

= 2ln 3 − 2ln 1

ln 1 is 0, because e to the 0 is 1.

ln 1 = 0

So the integral is 2 ln 3.

A calculator gives about 2.197.

= 2ln 3 ≈ 2.197

4 Check

Let's check with a calculator: it adds up 1000 thin strips.

That gives about 2.197, the same as our answer.

Right!

1000 thin strips: ≈ 2.197

Let's look back.

First: the rules.

Then: integrate.

Then: top minus bottom.

Last: check.

Answer