Evaluate ∫ from 2 to 6 of 3/x dx Step by Step

Step by step

The question

Question: find the integral from 2 to 6 of 3 over x.

∫ from 2 to 6 of 3x 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

3 times 1 over the bottom gives 3 ln|x|.

No + C: we work out top minus bottom.

∫ from 2 to 6 of 3x dx = [ 3ln|x| ] from 2 to 6

3 Top minus bottom

Put in 6, then take away the same with 2.

= 3ln 6 − 3ln 2

Two logs taken away make one log of a fraction.

ln a − ln b = ln ab

So the integral is 3 ln 3.

A calculator gives about 3.296.

= 3ln 3 ≈ 3.296

4 Check

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

That gives about 3.296, the same as our answer.

Right!

1000 thin strips: ≈ 3.296

Let's look back.

First: the rules.

Then: integrate.

Then: top minus bottom.

Last: check.

Answer