Step by step
The question
Question: find the integral from 0 to 1 of 1 over x.
1 Why is it improper?
Draw the curve.
At x = 0 the curve shoots up for ever: 1/0 has no value.
The area is not closed at the top.
We cannot put in 0.
So we start at a letter, t, just right of 0, and then let t shrink to 0.
2 Use a limit
The integral is the limit of the integral from t to 1, as t goes to 0 from the right.
3 Integrate
1/x integrates to ln x.
The power rule cannot do it.
4 Let t go
Put in 1, then take away the same with t.
As t shrinks to 0, ln t goes down without end, slowly but for ever.
5 Decide
The limit is infinity: the area has no end.
The integral diverges.
Careful!
Never put 0 in like a number: 1/0 has no value.
We let t shrink and watch where the answer goes.
6 Check
Let's check with a calculator: from t = 0.001 the integral is about 6.91.
From a millionth, about 13.82.
It keeps growing.
Right!
Let's look back.
First: why is it improper?.
Then: use a limit.
Then: integrate.
Then: let t go.
Then: decide.
Last: check.