Step by step
The question
Question: find the integral from 0 to infinity of e⁻²ˣ.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Why it is 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 0 to t, as t goes to infinity.
3 Integrate
e to the minus 2x integrates to itself over minus 2.
4 Let t go
Put in t, then take away the same with 0.
As t grows, e to the minus t is 1/a huge number: it goes to 0.
The first part goes to 0.
e to the 0 is 1.
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.
Up to a million, about 0.5.
Closer and closer to 0.5.
Right!
7 Pick the choice
Our answer is 1/2.
That is choice A.
8 Why not the others
B is −1/2.
Wrong: the sign at the bottom end.
C is infinity.
Wrong: e⁻²ˣ goes to 0 fast enough.
D is 2.
Wrong: divided the wrong way: ∫ e⁻²ˣ dx = −e⁻²ˣ/2.
Let's look back.
First: why it is improper.
Then: use a limit.
Then: integrate.
Then: let t go.
Then: decide.
Then: check.
Then: pick the choice.
Last: why not the others.