Evaluate ∫ from 0 to ∞ of e^(-2x) dx Step by Step

Step by step

The question

Question: find the integral from 0 to infinity of e⁻²ˣ.

∫ from 0 to ∞ of e^−2x 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 0 to t, as t goes to infinity.

∫ from 0 to ∞ of e^−2x dx = lim t → ∞ ∫ from 0 to t of e^−2x dx

3 Integrate

e to the minus 2x integrates to itself over minus 2.

= lim t → ∞ [ −12e^−2x ] from 0 to t

4 Let t go

Put in t, then take away the same with 0.

= lim t → ∞ (−12e^−2t + 12e⁰)

As t grows, e to the minus t is 1/a huge number: it goes to 0.

e^−t = 1e^t → 0

The first part goes to 0.

e to the 0 is 1.

= 0 + 12

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!

= 12

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 0.5.

Already close to 0.5.

Right!

t = 1000: 0.5

Let's look back.

First: why is it improper?.

Then: use a limit.

Then: integrate.

Then: let t go.

Then: decide.

Last: check.

Answer