Step by step
The question
Question: find the integral from 1 to infinity of 1/x³.
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.
2 Use a limit
The integral is the limit of the integral from 1 to t, as t goes to infinity.
3 Integrate
The power rule: x to the minus 3 becomes x to the −2, divided by −2.
4 Let t go
Put in t, then take away the same with 1.
As t grows, minus 1/(2t²) goes to 0.
The t part goes to 0.
5 Decide
The limit is a number: the integral converges to 1/2.
An area that goes on for ever, but adds up to 1/2!
Careful!
Never put infinity in like a number.
We let t grow and watch where the answer goes.
6 Check
Let's check with a calculator: the integral up to t = 1000 is about 0.5.
Already close to 0.5.
Right!
Let's look back.
First: why is it improper?.
Then: use a limit.
Then: integrate.
Then: let t go.
Then: decide.
Last: check.