Evaluate ∫ from 1 to ∞ of 1/√x dx Step by Step

Step by step

The question

Question: find the integral from 1 to infinity of 1 over √x.

∫ from 1 to ∞ of 1√x dx =

1 Why is it improper?

Draw the curve.

The area under it goes on to the right for ever: the top limit is infinity.

Infinity is not a number we can put in.

So we stop at a letter, t, and then let t grow without end.

∞ is not a number: use t, then let t → ∞

2 Use a limit

The integral is the limit of the integral from 1 to t, as t goes to infinity.

∫ from 1 to ∞ of 1√x dx = lim t → ∞ ∫ from 1 to t of 1√x dx

3 Integrate

The power rule: x to the minus 1/2 becomes x to the 1/2, divided by 1/2.

= lim t → ∞ [ 2√x ] from 1 to t

4 Let t go

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

= lim t → ∞ (2√t − (2))

As t grows, 2 √t grows without end.

2√t → ∞

5 Decide

The limit is infinity: the area has no end.

The integral diverges.

the limit = ∞

Careful!

The curve goes down to 0, but too slowly: the area still grows for ever.

Careful: not the curve → 0, so the area is a number ✗

6 Check

Let's check with a calculator: up to t = 1000 the integral is about 61.25.

t = 1000: 61.25

Up to a million, about 1998.

It keeps growing.

Right!

t = 1 000 000: 1998

Let's look back.

First: why is it improper?.

Then: use a limit.

Then: integrate.

Then: let t go.

Then: decide.

Last: check.

Answer