Which is ∫ e^(5x) dx?

Step by step

The question

Question: find the integral of e⁵ˣ.

Two new rules: one for e to a power, one for 1 over x.

There are four choices.

First we solve it, the normal way.

Then we pick.

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.

With e⁵ˣ: its derivative is 5 times e⁵ˣ, by the chain rule.

That 5 must not be there.

So we divide by 5.

Careful!

Do not forget to divide by 5.

2 Integrate

Divide by 5: we get 1/5 times e⁵ˣ.

Add C: the slope cannot tell us a number added on the end.

So the answer is 1/5 × e⁵ˣ + C.

3 Check

Let's check: take the derivative of our answer.

e to the kx gives k times e to the kx.

The numbers cancel, and we get e⁵ˣ: the question again.

Right!

4 Pick the choice

Our answer is 1/5e⁵ˣ + C.

That is choice C.

5 Why not the others

A is 5e⁵ˣ + C.

Wrong: that is the derivative.

B is e⁶ˣ/6 + C.

Wrong: the power rule does not work for e.

D is e⁵ˣ + C.

Wrong: did not divide by 5.

Let's look back.

First: the rules.

Then: integrate.

Then: check.

Then: pick the choice.

Last: why not the others.