Evaluate ∫ from 1 to e of x ln x dx Step by Step

Step by step

The question

Question: find the integral from 1 to e of x ln x.

∫ from 1 to e of x ln x dx =

1 Choose u and dv

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.

u = ln x, dv = x dx

2 Find du and v

The derivative of ln x is 1 over x.

And v is the integral of dv: 1/2 × x², by the power rule.

du = 1x dx, v = 12x²

3 Use the rule

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

It comes from the product rule, backwards.

∫ u dv = uv − ∫ v du

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

= 12x² ln x − ∫ 12x dx

4 The new integral

The new integral is easy: the power rule.

= 12x² ln x − 14x²

Careful!

It is u v minus the integral of v du.

Not plus!

Careful: not uv + ∫ v du ✗

Open the bracket.

No + C: top minus bottom.

= [ 12x² ln x − 14x² ] from 1 to e

5 Top minus bottom

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

= (12 × e² ln e − 14 × e²) − (12 × 1² ln 1 − 14 × 1²)

ln e is 1, and ln 1 is 0.

ln e = 1, ln 1 = 0

Tidy up: the integral is (e² + 1)/4, about 2.097.

= e² + 14 ≈ 2.097

6 Check

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

That gives about 2.097, the same as our answer.

Right!

1000 thin strips: ≈ 2.097

Let's look back.

First: choose u and dv.

Then: find du and v.

Then: use the rule.

Then: the new integral.

Then: top minus bottom.

Last: check.

Answer