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Thread: integration

  1. #1
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    integration

    integral(tan(inverse) (2*x-1)/(1+x-x^2)) is

    what should i do of tan(inverse) x is there any integration of it
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  2. #2
    MHF Contributor chisigma's Avatar
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    Re: integration

    With the substitution 1+x-x^{2}= u the integral becomes...

    -\int \tan^{-1} u\ du (1)

    Kind regards

    \chi \sigma
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  3. #3
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    Re: integration

    Quote Originally Posted by prasum View Post
    what should i do of tan(inverse) x is there any integration of it
    Yes - parts (after the sub given by chisigma).

    Just in case a picture helps...



    ... is lazy integration by parts, doing without u and v.

    ... where (key in spoiler) ...

    Spoiler:


    ... is the product rule, where straight continuous lines differentiating downwards with respect to x (or u, or whatever you've substituted for).


    So you would have...



    (but with u, or whatever; these I made earlier.)

    Spoiler:


    Spoiler:


    ... where...
    Spoiler:



    ... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to the main variable (x, or whatever), and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).





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