Which is ∫ x e^x dx? Step by Step | AP Calculus BC Practice Exam 1

Step by step

The question

Question: find the integral of x eˣ.

∫ xe^x dx

There are four choices.

First we solve it, the normal way.

Then we pick.

1 Choose u and dv

We will trade this integral for a new one with du in it.

So u should be the part that gets simpler when we take its derivative.

Choose u = x: its derivative is simpler, just a number.

dv is the rest: eˣ dx, which we can integrate.

u = x, dv = e^x dx

2 Find du and v

du: the derivative of u, times dx.

v: the integral of dv.

e to the x stays e to the x.

du = 1 dx, v = e^x

3 Use the rule

The rule: the integral of u dv is u v, minus the integral of v du.

∫ u dv = uv − ∫ v du

u times v, minus the integral of v times du.

= xe^x − ∫ e^x dx

4 The new integral

The new integral is easy: just a number times e to a power.

= xe^x − e^x

Careful!

It is u v minus the integral of v du.

Not plus!

Careful: not uv + ∫ v du ✗

Add C.

So the answer is x eˣ − eˣ + C.

= xe^x − e^x + C

5 Check

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

The product rule on each term: the first's slope times the second, plus the first times the second's slope.

derivative: e^x + xe^x − e^x

The other terms cancel, and we get x eˣ: the question again.

Right!

= xe^x

6 Pick the choice

Our answer is xeˣ − eˣ + C.

That is choice D.

D) xe^x − e^x + C

7 Why not the others

A is xeˣ + eˣ + C.

Wrong: the sign: minus ∫ v du.

That is the Careful mistake from before.

A) xe^x + e^x + C ✗ the sign: minus ∫ v du

B is eˣ + C.

Wrong: that is ∫ eˣ dx only.

B) e^x + C ✗ that is ∫ e^x dx only

C is x²eˣ/2 + C.

Wrong: integrated x only: use parts.

C) x²e^x/2 + C ✗ integrated x only: use parts

Let's look back.

First: choose u and dv.

Then: find du and v.

Then: use the rule.

Then: the new integral.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer