# Integrate improper integral with infinite discontenuities

1. Oct 30, 2012

### alust92

1. The problem statement, all variables and given/known data

Integrate the improper integral (use correct notation). State whether it's converging or diverging.
10
∫ 7/(x-9)^2 dx
8
2. Relevant equations

b c
∫ f(x) dx= lim ∫ f(x) dx
a c → d a

3. The attempt at a solution

b
lim ∫ 7(x-9)^2 dx
c → 10- 8

Let u= x-9
-du= dx

lim ∫ -u^2
c→10-

lim -2u^2
c→10-

lim -2(x-9)^2
c→10-

lim -2(10-9)^2-2(8-9)^2
c→10-

I know I must be doing something wrong because the answer is ∞, any ideas where I went wrong?

2. Oct 30, 2012

### Staff: Mentor

Mod note: Added [ code ] tags to preserve the OP's spacing.
The problem is not at 10 - it's at 9, which is where the denominator becomes zero. You're going to have to split the interval [8, 10] into two parts, and then deal with each separately.

3. Oct 30, 2012

### alust92

Ok, so here's what I've got.
9 10
∫ 7/(x-9)^2 dx + ∫ 7/(x-9)^2
8 9

9 9
∫ 7/(x-9)^2 dx = lim ∫ 7/(x-9)^2 dx
8 x→9 8

10 9
∫ 7/(x-9)^2 dx = lim ∫ 7/(x-9)^2 dx
8 x→9 8

10
∫ 7/(x-9)^2 dx
8

Am I on the right track?

Last edited: Oct 30, 2012
4. Oct 30, 2012

### Staff: Mentor

The above is correct.
The limit should be taken as b →9-, not as x →9. The upper limit of integration should be b.

What you have just below is just the repeat of the line above.