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Math Help - Int 1

  1. #1
    Boo
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    Int 1

    All!
    Please, can someone help me with:

    \int\frac{x-3}{x^2+1}dx

    The result should be:
    \frac{1}{2}ln|x^2+1|-3arctanx+C

    Many thanks!
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  2. #2
    Member sbhatnagar's Avatar
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    Re: Int 1

    Quote Originally Posted by Boo View Post
    All!
    Please, can someone help me with:

    \int\frac{x-3}{x^2+1}dx

    The result should be:
    \frac{1}{2}ln|x^2+1|-3arctanx+C

    Many thanks!
    Write this as

    \int\frac{x}{x^2+1}dx-3\int \frac{1}{1+x^2}dx

    For the first integral, substitute u=1+x^2.

    \int \frac{x}{x^2+1}dx = \frac{1}{2}\int \frac{1}{u}du

    =\frac{1}{2}\ln |u|+\text{Constant} = \frac{1}{2}\ln |1+x^2| +\text{Constant}

    For the second integral, substitute x=\tan \theta.

    \int \frac{1}{1+x^2}dx = \int \frac{\sec^2 \theta}{1+\tan^2 \theta}d\theta

    Note that 1+\tan^2 \theta =\sec^2 \theta, so our integral will become

    \int d\theta = \theta + \text{Constant} = \arctan(x)+\text{Constant}

    Combine these two results to obtain

    \int\frac{x}{x^2+1}dx-3\int \frac{1}{1+x^2}dx

    = \frac{1}{2}\ln|1+x^2|-3\arctan(x)+\text{Constant}
    Thanks from Boo
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  3. #3
    Boo
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    Re: Int 1

    Many thanks, sb!!!
    please, how did U get the second line? What is the rule?
    Many thanks!!!
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  4. #4
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    Re: Int 1

    Have you never taken a Calculus class? What sbhatnagar did is pretty standard. To integrate \int \frac{x}{x^2+ 1}dx, seeing that " x^2" in the denominator and the "xdx" in the numerator, you should immediately think to try the substitution u= x^2+ 1 so that du= 2xdx or \frac{1}{2}du= x dx. We can write
    \int \frac{x dx}{x^2+ 1}= \int\frac{\frac{1}{2}du}{u}= \frac{1}{2}\int \frac{du}{u}.

    And now you should recognise that \frac{d ln(u)}{du}= \frac{1}{u} so that \int \frac{du}{u}= ln|u|+ C.
    (Note the absolute value: ln|u|. ln(u) is only defined for postive u and u, here, might be negative.)
    Thanks from Boo
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  5. #5
    Boo
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    Re: Int 1

    Thank You , Hallsofivy, that was really helpful!!!AM really new here!
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  6. #6
    Boo
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    Re: Int 1

    Please, how did he get this with x_=tan fi??
    many thanks!
    p.s. trying to google it but it is not that easy...if someone has good pleages please send it...am new here...MANY THANKS!!!
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  7. #7
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    Re: Int 1

    Just in case a picture helps...



    That's how the substitution works. However, you can see that it depends on the chain rule, and I would prefer to see how we are working backwards through a differentiation according to the chain rule.

    If, like me, you were taught to see differentiation as a downwards journey and integration up, then you'll see the logic of this...



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

    Spoiler:


    ... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to the main variable (in this case x), 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).

    The general drift is...





    So, the bottom row shows that we have multiplied the derivative with respect to the inner function by the derivative of the inner function itself. Which is what the chain rule wants.

    However, the inner function is just x, so the derivative of that is just one, so no big deal from applying the chain rule.

    But now, carefully, remove the inner function (by twisting gently) and replace with tan theta...



    Now the derivative of the inner function is sec squared so, as sbhatnagar explains, we are really integrating (or anti-differentiating) one, with respect to theta...



    Then just replace the original inner function...





    _________________________________________

    Don't integrate - balloontegrate!

    Balloon Calculus; standard integrals, derivatives and methods

    Balloon Calculus Drawing with LaTeX and Asymptote!
    Last edited by tom@ballooncalculus; January 20th 2013 at 12:26 PM.
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  8. #8
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    Re: Int 1

    The first integral is of the other main kind (that involve a substitution). It comes ready with an appropriate inner function...



    So you only need to deduce that the function F is ln...





    _________________________________________

    Don't integrate - balloontegrate!

    Balloon Calculus; standard integrals, derivatives and methods

    Balloon Calculus Drawing with LaTeX and Asymptote!
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