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Math Help - Steps in integration problem

  1. #1
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    Steps in integration problem

    Hi, while working on some integration problems today, I came across this one:

    Steps in integration problem-picture-13.png
    I multiplied it out:

    Steps in integration problem-picture-15.png
    (Edit: it's supposed to say x^2+2x in the first term)

    And with some trial and error I got an answer:

    Steps in integration problem-picture-14.png
    When I differentiate this I get back what I started out with.

    But trial and error isn't usually the best way to get an answer, so can anyone tell me which steps I need to take here?

    Should I be using U-substitution?
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  2. #2
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    if t=x^2+2x\implies dt=(2x+2)\,dx=2(x+1)\,dx.
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  3. #3
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    Quote Originally Posted by TwoPlusTwo View Post
    Hi, while working on some integration problems today, I came across this one:

    Click image for larger version. 

Name:	Picture 13.png 
Views:	18 
Size:	6.1 KB 
ID:	19466
    I multiplied it out:

    Click image for larger version. 

Name:	Picture 15.png 
Views:	18 
Size:	6.4 KB 
ID:	19468
    (Edit: it's supposed to say x^2+2x in the first term)

    And with some trial and error I got an answer:

    Click image for larger version. 

Name:	Picture 14.png 
Views:	18 
Size:	5.3 KB 
ID:	19467
    When I differentiate this I get back what I started out with.

    But trial and error isn't usually the best way to get an answer, so can anyone tell me which steps I need to take here?

    Should I be using U-substitution?


    You can do that, or else you can notice that x+1 is half the derivative of x^2+2x , so you can write

    \int (x+1)e^{x^2+2x}dx = \frac{1}{2}\int (2x+2)e^{x^2+2x}dx and then use the almost-immediate integral

    \int f'(x)e^{f(x)}dx=e^{f(x)}+C

    Tonio
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  4. #4
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    Makes perfect sense. Thanks guys! I had a feeling I was missing something obvious.
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