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

Step by step

The question

Question: find the integral from 1 to infinity of 1/x².

∫ from 1 to ∞ of 1x² 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 1x² dx = lim t → ∞ ∫ from 1 to t of 1x² dx

3 Integrate

The power rule: x to the minus 2 becomes x to the −1, divided by −1.

= lim t → ∞ [ −1x ] from 1 to t

4 Let t go

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

= lim t → ∞ (−1t − (−1))

As t grows, minus 1/t goes to 0.

−1t → 0

The t part goes to 0.

= 0 − (−1)

5 Decide

The limit is a number: the integral converges to 1.

An area that goes on for ever, but adds up to 1!

= 1

Careful!

Never put infinity in like a number.

We let t grow and watch where the answer goes.

Careful: not put in ∞: −1∞ ✗

6 Check

Let's check with a calculator: the integral up to t = 1000 is about 0.999.

Already close to 1.

Right!

t = 1000: 0.999

Let's look back.

First: why is it improper?.

Then: use a limit.

Then: integrate.

Then: let t go.

Then: decide.

Last: check.

Answer