# How do you evaluate the integral of #int (dt)/(t-4)^2# from 1 to 5?

##### 3 Answers

Substitute

Answer is, if you are indeed asked to just find the integral:

If you seek the area, it's not that simple though.

#### Explanation:

Set:

Therefore the differential:

And the limits:

Now substitute these three values found:

NOTE: **DO NOT READ THIS IF YOU HAVEN'T BEEN TAUGHT HOW TO FIND THE AREA**. Although this should actually represent the area between the two limits and since it is always positive, it should have been positive. However, this function is **not continuous** at

#### Explanation:

Depending on how much integration you've learned the "best" answer will be either: "the integral is not defined" (yet) **or** "the integral diverges"

#### Explanation:

When we try to evaluate

**not** defined on the whole interval

**Early in the study of calculus** , we define the integral by starting with

"Let

#f# be define on interval#[a,b]# . . . "

So early in our study, the best answer is that

*is not defined* (yet?)

**Later we extend the definition** to what are called "improper integrals"

These include integrals on unbounded intervals (

To (try) to evaluate

(Note that the integrand is still not defined on these *closed* intervals.)

The method is to replace the point where the integrand is undefined by a variable, then take a limit as that variable approaches the number.

Let's find the integral first:

# = (-1/(b-4))-(-1/(-3))#

# = -1/(b-4)-1/3#

Looking for the limit as

Therefore the integral over

We say that the integral diverges.

**Note**

Some would say: we now have a *definition* of the integral, there just doesn't happen to be any number that satisfies the definition.