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George Straight

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8 Ap calculas problems

I'm having trouble with some math problems help on any of them is appreciated. Thanks!

Find the indefinite integral of these.

1. 2/(x-11)^6 dx     2. (X^3)(e^(3x^2)) dx  3. (2(lnx)^2)/(x^2) dx  4. xln(x+7) dx

These two I need to evaluate the limit and I just learned L'hopital's rule.
These are pretty hard i thought.

1. lim x->0+     (7(e^(x)-1-x)/(5x^3)

2. lim x->0+     (ln(x^8))/(x^5)

Evaluate the integral. These both converge so, they're do able.

1. from 0 to Infinity of xe^(-x/11)

2. from negative infinity to positive infinity of 10/(144+x^2)

Thanks again for any help on any of these!

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: 8 Ap calculas problems

DISCLAIMER: I have no paper and so have only been able to do these in my head. They could be wrong, or are not the best methods to use, so don't shoot me.

1. Rewrite as

. The outside is a multiple of the derivative of the inside - so add 1 to the power and adjust the constant (i.e. differentiate and see by what factor you are out).

2. Try

as a substitution.

3. Rewrite as

. Use integration by parts:

4. Use parts:

, but when you come to integrate, looks like you'll need to use

as a substitution.

Last edited by Daniel123 (2008-05-16 01:10:40)

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: 8 Ap calculas problems

I have never been formally taught limits, so I am doing these with bits of knowledge I have picked up - hopefully someone will tell me if I'm wrong.

Pretty much the only way I know of evaluating limits is L'Hopital's, so here goes:

1. The limits of both the top and the bottom as x tends to 0 are 0, so this *I think* qualifies the use of L'Hopital's (using it twice):

2. I'm not sure about this. Graphing it shows that the limits are different as x approaches 0 from both sides. What does this mean? darn British calculus.

Last edited by Daniel123 (2008-05-15 10:06:37)

George Straight

Guest

Re: 8 Ap calculas problems

You're supposed to approach it from the right that's why it's 0+ oppositely it could be 0- sorry for the confusion.:D

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: 8 Ap calculas problems

Daniel is saying that the numerator and denominator have different limits, and so L'Hopital doesn't apply.

Evaluating that one directly shows that it goes to -∞.

---

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It wanted to be normal.

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: 8 Ap calculas problems

Ok for your last two:

1. Use parts?

2.

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: 8 Ap calculas problems

Argh. Turns out this can be done much more quickly:

I haven't been taught this yet though, so I have an excuse

Edit : And I've just constructed a little proof of it Generalising the substitution I made in my earlier post:

Then going back to the initial substitution:

Last edited by Daniel123 (2008-05-16 01:00:44)