Evaluate ∫ from 1 to ∞ of 1/x dx Step by Step

Step by step

The question

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

∫ from 1 to ∞ of 1x dx =

1 Why is it improper?

Draw the curve.

The area under it goes on to the right for ever: the top limit is infinity.

Infinity is not a number we can put in.

So we stop at a letter, t, and then let t grow without end.

∞ is not a number: use t, then let t → ∞

2 Use a limit

The integral is the limit of the integral from 1 to t, as t goes to infinity.

∫ from 1 to ∞ of 1x dx = lim t → ∞ ∫ from 1 to t of 1x dx

3 Integrate

1/x integrates to ln x.

The power rule cannot do it.

= lim t → ∞ [ ln x ] from 1 to t

4 Let t go

Put in t, then take away the same with 1: ln 1 is 0.

= lim t → ∞ ( ln t − (0))

As t grows, ln t grows without end, slowly but for ever.

ln t → ∞

5 Decide

The limit is infinity: the area has no end.

The integral diverges.

the limit = ∞

Careful!

The curve goes down to 0, but too slowly: the area still grows for ever.

Careful: not the curve → 0, so the area is a number ✗

6 Check

Let's check with a calculator: up to t = 1000 the integral is about 6.91.

t = 1000: 6.91

Up to a million, about 13.82.

It keeps growing.

Right!

t = 1 000 000: 13.82

Let's look back.

First: why is it improper?.

Then: use a limit.

Then: integrate.

Then: let t go.

Then: decide.

Last: check.

Answer