Convergence Divergence Help

cooltee13

1. The problem statement, all variables and given/known data
Well im studying for my final which is in a couple of days, and Im stuck on this topic of convergence of improper integrals. Ive been doing a couple and I thought I was doing them correctly but my answers do not go with the answers I was given. So Im stressing out over it. Here are the questions I have:

Determine whether the following integrals converge or diverge, evaluate those that converge.

1) Integral from -inf to 1 of: dx/(x-2)^5

2) Integral from 0 to inf of: e^x dx


2. Relevant equations
Just the standard limit and integral equations


3. The attempt at a solution
1) I split the integral into two pieces, a) the int from -inf to 0: 1/(x-2)^5 and b) the int from 0 to 1: 1/(x-2)^5 I then took the limit of each integral, so I got:
a) lim as t approaches -inf of a. solving the limit I got (1/-2 - 1/inf) which = -1/2 and is convergant.

b) lim as t approaches 0 of b. solving that limit I came up with a divergent answer. making the original Integral divergent. however the answer I was given says that the integral is convergent. Im confused on this.
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2) integral from 0 to inf of e^x dx
I split this integral into two pieces as well. a) integral from 0 to 1: e^x dx and b) integral from 1 to inf: e^x dx

Solving each part, I got two convergent answers, but the answer said that its divergent. Im wrong again .

Please help me out, and explain why. I have the answers, thats not what im going for..I just want to understand this so I can do well on my final. Thanks guys

rootX

I didn't get why you are splitting things.
You only do that when you have a discontinuity like going from -1 to 1 for 1/x.

1/(x-2)^2 .. gives -1/(x-2) ] from t to 1, and take limit of t as it appraches -inf

same thing for e^x

rootX

also, you did this step wrong!
you don't have any inf/discountinity from 0 to 1