Which is ∫ x^2 ln x dx, for x

Step by step

The question

Question: find the integral of x² ln x.

∫ x² ln 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.

Choose u = ln x: its derivative, 1/x, is simpler.

We do not know how to integrate ln x yet.

So dv is the rest: x² dx.

u = ln x, dv = x² dx

2 Find du and v

The derivative of ln x is 1 over x.

And v is the integral of dv: 1/3 × x³, by the power rule.

du = 1x dx, v = 13x³

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.

= 13x³ ln x − ∫ 13x² dx

4 The new integral

The new integral is easy: the power rule.

= 13x³ ln x − 19x³

Careful!

It is u v minus the integral of v du.

Not plus!

Careful: not uv + ∫ v du ✗

Add C.

So the answer is 1/3 × x³ ln x − 1/9 × x³ + C.

= 13x³ ln x − 19x³ + 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: x² ln x + 13x² − 13x²

The other terms cancel, and we get x² ln x: the question again.

Right!

= x² ln x

6 Pick the choice

Our answer is 1/3x³ ln x − 1/9x³ + C.

That is choice D.

D) x³/3 ln x − x³/9 + C

7 Why not the others

A is x³ over 3 ln x + C.

Wrong: left out the ∫ v du part.

A) x³/3 ln x + C ✗ left out the ∫ v du part

B is x³ over 3 ln x + x³ over 9 + C.

Wrong: the sign: minus ∫ v du.

That is the Careful mistake from before.

B) x³/3 ln x + x³/9 + C ✗ the sign: minus ∫ v du

C is 2x ln x + x + C.

Wrong: differentiated instead.

C) 2x ln x + x + C ✗ differentiated instead

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