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Thread: Integral by substitution

  1. #1
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    Integral by substitution

    Hi, I have the following problem but I do not know what to do...what substitution should I choose? Thanks in advance!

    Using a substitution of the form $\displaystyle y=a+b sinh(t)$, or otherwise, find polynomial $\displaystyle p(y)$ such that $\displaystyle \int \sqrt{10+y^2-2y} dy = +p(y) \sqrt{10+y^2-2y} + C$.
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  2. #2
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    What does the plus that precedes p(y) indicate? If you want the integral:

    Write $\displaystyle y^2-2y+10 = (y-1)^2+9$ and let $\displaystyle y+1 = 3\tan{t}$.
    You will need to find the integral $\displaystyle \sec^3{t}$, for which you should see here.
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  3. #3
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    Help, please...

    Hi,

    I am not sure...I guess it means that it is positive? It does not say anything else.

    I have done it but I have got the wrong result, I think.

    I got
    $\displaystyle 9/2 arsinh((y-1)/3)+(1/2)((y-1)/3) \sqrt{10+y^2+2y} + C$

    I used a different substitution though, $\displaystyle y=1+3 sinh(t)$, with $\displaystyle dy=3 cosh(t) dt$.

    A summary of my calculations is here:

    $\displaystyle \int \sqrt{10+y^2-2y} dy = \int \sqrt{(y-1)^2+3^2} dy$
    $\displaystyle = \int \sqrt{(3 sinh(t))^2+3^2} 3 cosh(t) dt $
    $\displaystyle = 9 \int \sqrt{1+sinh^2(t)} cosh(t) dt = 9 \int cosh^2(t) dt $
    $\displaystyle = 9/2 \int (1 + cosh(2t)) dt = 9/2 (t + sinh(2t)/2) + C = 9/2 (t sinh(t) + cosh(t)) + C$

    Substituting back to y:

    $\displaystyle = 9/2 * arsinh((y-1)/3) + 1/2 * ((y-1)/3) \sqrt{10+y^2-2y} + C$

    (using the fact that $\displaystyle cosh(t) = \sqrt{1+sinh^2(t)} = \sqrt{10+y^2-2y}$
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  4. #4
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    Quote Originally Posted by juanma101285 View Post
    Hi, I have the following problem but I do not know what to do...what substitution should I choose? Thanks in advance!

    Using a substitution of the form $\displaystyle y=a+b sinh(t)$, or otherwise, find polynomial $\displaystyle p(y)$ such that $\displaystyle \int \sqrt{10+y^2-2y} dy = +p(y) \sqrt{10+y^2-2y} + C$.
    $\displaystyle \displaystyle \sqrt{y^2 - 2y + 10} = \sqrt{y^2 - 2y + (-1)^2 - (-1)^2 + 10}$

    $\displaystyle \displaystyle = \sqrt{(y-1)^2 + 9}$.


    Now make the substitution $\displaystyle \displaystyle y - 1 = 3\sinh{t}$ so that $\displaystyle \displaystyle dy = 3\cosh{t}\,dt$.


    Then your integral becomes

    $\displaystyle \displaystyle \int{\sqrt{(y-1)^2 + 9}\,dy} = \int{\sqrt{(3\sinh{t})^2 + 9}\,3\cosh{t}\,dt}$

    $\displaystyle \displaystyle = \int{3\cosh{t}\sqrt{9\sinh^2{t} + 9}\,dt}$

    $\displaystyle \displaystyle = \int{3\cosh{t}\sqrt{9(\sinh^2{t} + 1)}\,dt}$

    $\displaystyle \displaystyle = \int{3\cosh{t}\sqrt{9\cosh^2{t}}\,dt}$

    $\displaystyle \displaystyle = \int{3\cosh{t}\cdot 3\cosh{t}\,dt}$

    $\displaystyle \displaystyle = \int{9\cosh^2{t}\,dt}$

    $\displaystyle \displaystyle = \int{9\left(\frac{1}{2}\cosh{2t} + \frac{1}{2}\right)\,dt}$

    $\displaystyle \displaystyle = \int{\frac{9}{2}\cosh{2t} + \frac{9}{2}\,dt}$

    $\displaystyle \displaystyle = \frac{9}{4}\sinh{2t} + \frac{9}{2}t + C$

    $\displaystyle \displaystyle = \frac{9}{2}\sinh{t}\cosh{t} + \frac{9}{2}t + C$

    $\displaystyle \displaystyle = \frac{9}{2}\sinh{t}\sqrt{1 + \sinh^2{t}} + \frac{9}{2}t + C$

    $\displaystyle \displaystyle = \frac{9}{2}\left(\frac{y-1}{3}\right)\sqrt{1 + \left(\frac{y-1}{3}\right)^2} + \frac{9}{2}\sinh^{-1}\left(\frac{y-1}{3}\right) + C$

    $\displaystyle \displaystyle = \frac{3(y-1)}{2}\frac{\sqrt{y^2 - 2y + 10}}{\sqrt{9}} + \frac{9}{2}\sinh^{-1}\left(\frac{y-1}{3}\right) + C$

    $\displaystyle \displaystyle = \frac{y-1}{2}\sqrt{y^2 - 2y + 10} + \frac{9}{2}\sinh^{-1}\left(\frac{y-1}{3}\right) + C$.
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    Thanks!
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