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Integration by sketching

  1. Sep 7, 2015 #1
    I greatly appreciate this chance to submit a query.

    I have the following integral: $$\int_{1}^t 2sin(t-\tau)e^{-2(t-1)} d\tau$$

    and it has been suggested to me that if I sketch the two constituent functions and multiply them, I can read the answer off the paper. So here are my sketches: go straight to the 3rd arrow


    Don't let alternative integral side-track you; it involves something called "convolution" and is something I am grappling with too.

    My firtst attempt: There is clearly some area under the graphs, so the answer is not 0. The alternatives I have are: 0, 1 or 2. But, surely, that's not something I can read that off the graph?

    My second attempt: I just did the integral by taking $$2e^{-2(t-1)}$$ out; but that simply integrates down to an expression in terms of 't', and without any value of 't' I can't see how a definite answer can be obtained.

    Any assistance with tackling the above would be helpful I am sure. Your advice is sought.

    Best regards,
  2. jcsd
  3. Sep 7, 2015 #2


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    Gold Member

    I think you can put outside the integral the factors without ##\tau## so you can integrate ##\int_{1}^{t}\sin{(t-\tau)}d\,\tau## that is elementary, you obtain in this way an Area function that depends only by ##t##.
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