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Math Help - Improper Integrals Help

  1. #1
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    Unhappy Improper Integrals Help

    Uh I've posted a couple times before on improper integrals but this time I'm just going to ask all the questions and kind of scenarios that I have. If you can help me with these questions the other questions in my textbook will be easier because its pretty much the same from there.

    1) x e^-x^2 dx between the interval of infinity and -infinity

    For this question, I saw an example of it on pauls notes and well so I see that when dealing with interval between infinities means that you have to split the integral up, am i correct? Thats my question for this.

    2) x+1/x^2+2x dx between infinity and 1

    For this question how would I integrate this, I forgot how to integrate fractions well I know that say x/2x+3x I think would be integrated by first doing x/2x+x/3x am I right? But I'm not sure how how its done when it has two variables at the top and bottom.

    3) S e^-5x ds between infinity and 0

    For this question I've seen the answer for it and it tells me to use integration by parts. Why do I have to do integration by parts for this, isn't S just a variable aren't I able to keep the S out and just integrate the e^-5x and solve? How would I do this question or attempt it?

    4) ln x/x dx between infinity and t

    For this question I don't know how to approach it all.

    5) x2/9+x^6 dx between infinity and -infinity

    Lastly for this question since its between the infinities im guessing you have to split them up but again I'm not sure how since I don't know how to integrate fractions such as these.

    Thanks in advance guys, I really need your help
    Sorry if i'm being dumb but I need all the steps i can take for these questions :P
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  2. #2
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    Re: Improper Integrals Help

    Quote Originally Posted by kashmoneyrecord3 View Post
    5) x2/9+x^6 dx between infinity and -infinity
    Here is a hint on this one.
    What is the derivative of \frac{1}{9}\arctan \left( {\frac{{x^3 }}{3}} \right)~?
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  3. #3
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    Re: Improper Integrals Help

    uh i'm not sure what is it? help please
    Last edited by kashmoneyrecord3; October 15th 2011 at 04:26 PM.
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    Re: Improper Integrals Help

    Quote Originally Posted by kashmoneyrecord3 View Post
    uh i'm not sure what is it? help please
    If you cannot do derivatives, the how in the world do you think you can do anti-derivatives?
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  5. #5
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    Re: Improper Integrals Help

    Quote Originally Posted by kashmoneyrecord3 View Post
    Uh I've posted a couple times before on improper integrals but this time I'm just going to ask all the questions and kind of scenarios that I have. If you can help me with these questions the other questions in my textbook will be easier because its pretty much the same from there.

    1) x e^-x^2 dx between the interval of infinity and -infinity

    For this question, I saw an example of it on pauls notes and well so I see that when dealing with interval between infinities means that you have to split the integral up, am i correct? Thats my question for this.

    2) x+1/x^2+2x dx between infinity and 1

    For this question how would I integrate this, I forgot how to integrate fractions well I know that say x/2x+3x I think would be integrated by first doing x/2x+x/3x am I right? But I'm not sure how how its done when it has two variables at the top and bottom.

    3) S e^-5x ds between infinity and 0

    For this question I've seen the answer for it and it tells me to use integration by parts. Why do I have to do integration by parts for this, isn't S just a variable aren't I able to keep the S out and just integrate the e^-5x and solve? How would I do this question or attempt it?

    4) ln x/x dx between infinity and t

    For this question I don't know how to approach it all.

    5) x2/9+x^6 dx between infinity and -infinity

    Lastly for this question since its between the infinities im guessing you have to split them up but again I'm not sure how since I don't know how to integrate fractions such as these.

    Thanks in advance guys, I really need your help
    Sorry if i'm being dumb but I need all the steps i can take for these questions :P
    1. \displaystyle f(x) = x\,e^{-x^2} and \displaystyle f(-x) = -x\,e^{-(-x^2)} = -x\,e^{-x^2} = -f(x).

    Since \displaystyle f(-x) = -f(x), we have an odd function.

    What do you know about \displaystyle \int_{-a}^a{f(x)\,dx} if \displaystyle f(x) is an odd function?


    2.

    \displaystyle \int{\frac{x+1}{x^2+2x}\,dx} = \frac{1}{2}\int{\frac{2x+2}{x^2+2x}\,dx}

    Now make the substitution \displaystyle u = x^2 + 2x \implies du = 2x + 2.


    3. If S is a constant, then you can take it out of the integral. If S is a variable, then you have a product of functions where a substitution is not appropriate, so integration by parts would be the way to go.


    4. Try making the substitution \displaystyle u = \ln{x} \implies du = \frac{1}{x}\,dx.


    5. If \displaystyle f(x) = \frac{x^2}{9 + x^6} then \displaystyle f(-x) = \frac{(-x)^2}{9 + (-x)^6} = \frac{x^2}{9 + x^6} = f(x).

    Since \displaystyle f(x) = f(-x) we have an even function. What do you know about integrals of the form \displaystyle \int_{-a}^a{f(x)\,dx} if \displaystyle f(x) is an even function?
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  6. #6
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    Re: Improper Integrals Help

    Quote Originally Posted by Prove It View Post
    1. \displaystyle f(x) = x\,e^{-x^2} and \displaystyle f(-x) = -x\,e^{-(-x^2)} = -x\,e^{-x^2} = -f(x).

    Since \displaystyle f(-x) = -f(x), we have an odd function.

    What do you know about \displaystyle \int_{-a}^a{f(x)\,dx} if \displaystyle f(x) is an odd function?


    2.

    \displaystyle \int{\frac{x+1}{x^2+2x}\,dx} = \frac{1}{2}\int{\frac{2x+2}{x^2+2x}\,dx}

    Now make the substitution \displaystyle u = x^2 + 2x \implies du = 2x + 2.


    3. If S is a constant, then you can take it out of the integral. If S is a variable, then you have a product of functions where a substitution is not appropriate, so integration by parts would be the way to go.


    4. Try making the substitution \displaystyle u = \ln{x} \implies du = \frac{1}{x}\,dx.


    5. If \displaystyle f(x) = \frac{x^2}{9 + x^6} then \displaystyle f(-x) = \frac{(-x)^2}{9 + (-x)^6} = \frac{x^2}{9 + x^6} = f(x).

    Since \displaystyle f(x) = f(-x) we have an even function. What do you know about integrals of the form \displaystyle \int_{-a}^a{f(x)\,dx} if \displaystyle f(x) is an even function?
    Please don't post more than two questions in a thread. Otherwise the thread can get convoluted and difficult to follow. See rule #8: http://www.mathhelpforum.com/math-he...hp?do=vsarules.

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