Evaluate ∫ from 0 to 1 of e^(2x) dx Step by Step

Step by step

The question

Question: find the integral from 0 to 1 of e²ˣ.

∫ from 0 to 1 of e^2x dx =

1 The rules

First, e to the x.

Its derivative is eˣ itself: its slope equals its height.

So its integral is eˣ + C too.

∫ e^x dx = e^x + C

With e²ˣ: its derivative is 2 times e²ˣ, by the chain rule.

That 2 must not be there.

So we divide by 2.

∫ e^2x dx = 12e^2x + C

Careful!

The derivative of e²ˣ is 2 times e²ˣ.

So e²ˣ alone is not the answer: divide by 2.

Careful: not ∫ e^2x dx = e^2x ✗

2 Integrate

Divide by 2: we get 1/2 times e²ˣ.

No + C: we work out top minus bottom.

∫ from 0 to 1 of e^2x dx = [ 12e^2x ] from 0 to 1

3 Top minus bottom

Put in 1, then take away the same with 0.

The 1/2 in front times both.

= 12(e² − e⁰)

e to the 0 is 1.

e⁰ = 1

So the integral is 1/2 times (e² − 1).

A calculator gives about 3.195.

= 12(e² − 1) ≈ 3.195

4 Check

Let's check with a calculator: it adds up 1000 thin strips.

That gives about 3.195, the same as our answer.

Right!

1000 thin strips: ≈ 3.195

Let's look back.

First: the rules.

Then: integrate.

Then: top minus bottom.

Last: check.

Answer