Integrate improper integral with infinite discontenuities

[alust92]

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Homework Statement

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

Homework Equations

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

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?

 

[Mark44]

Mod note: Added [ code ] tags to preserve the OP's spacing.[/color]

→

Homework Statement

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

Homework Equations

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

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?

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.

 

[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→9 810 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?

 

[Mark44]

Ok, so here's what I've got.

9                            10
∫   7/(x-9)^2 dx + ∫     7/(x-9)^2
8                           9

The above is correct.

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

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.

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?