Evaluate ∫ from 0 to 1 of 6e^(3x) dx Step by Step

Step by step

The question

Question: find the integral from 0 to 1 of 6e³ˣ.

∫ from 0 to 1 of 6e^3x 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 e³ˣ: its derivative is 3 times e³ˣ, by the chain rule.

That 3 must not be there.

So we divide by 3.

∫ e^3x dx = 13e^3x + C

Careful!

The derivative of e³ˣ is 3 times e³ˣ.

So e³ˣ alone is not the answer: divide by 3.

Careful: not ∫ e^3x dx = e^3x ✗

2 Integrate

Divide by 3: 6/3 is 2.

No + C: we work out top minus bottom.

∫ from 0 to 1 of 6e^3x dx = [ 2e^3x ] from 0 to 1

3 Top minus bottom

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

The 2 in front times both.

= 2(e³ − e⁰)

e to the 0 is 1.

e⁰ = 1

So the integral is 2 times (e³ − 1).

A calculator gives about 38.171.

= 2(e³ − 1) ≈ 38.171

4 Check

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

That gives about 38.171, the same as our answer.

Right!

1000 thin strips: ≈ 38.171

Let's look back.

First: the rules.

Then: integrate.

Then: top minus bottom.

Last: check.

Answer