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Math Help - Trig Substitution Problems

  1. #1
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    Trig Substitution Problems

    I have 3 problems which I can't really wrap my head around. I guess the ones I've been doing so far were much more literal. I was wondering if someone could help me out


    1.






    2.







    3. A charged rod of length L produces an electric field at point P(a,b) given by the integral below, where λ is the charge density per unit length on the rod and ε0 is the free space permittivity (see the figure) and b ≠ 0. Evaluate the integral to determine an expression for the electric field E(P). (Use lambda for λ and Emf_0 for ε0.)






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    Quote Originally Posted by elven06 View Post
    I have 3 problems which I can't really wrap my head around. I guess the ones I've been doing so far were much more literal. I was wondering if someone could help me out


    1.






    2.







    3. A charged rod of length L produces an electric field at point P(a,b) given by the integral below, where λ is the charge density per unit length on the rod and ε0 is the free space permittivity (see the figure) and b ≠ 0. Evaluate the integral to determine an expression for the electric field E(P). (Use lambda for λ and Emf_0 for ε0.)






    id help out but after trying the first one and seeing how many steps it'd take, and noticing that i still didnt match wolfram...very close to it though (so many steps!), makes me have to pass. i got quite far using integration by parts so that if \int f(x)g(x)dx is equal to your first intergral, set f(x)=1, and take its antiderivative in the I.B.P framework. very ugly and pointless integrals.
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    Quote Originally Posted by yoman360 View Post
    The first one is correct, the second isn't. Anyone else have any input they could provide?

    Thanks
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  5. #5
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    Quote Originally Posted by elven06 View Post
    I have 3 problems which I can't really wrap my head around. I guess the ones I've been doing so far were much more literal. I was wondering if someone could help me out


    1.






    2.







    3. A charged rod of length L produces an electric field at point P(a,b) given by the integral below, where λ is the charge density per unit length on the rod and ε0 is the free space permittivity (see the figure) and b ≠ 0. Evaluate the integral to determine an expression for the electric field E(P). (Use lambda for λ and Emf_0 for ε0.)







    For 1 & 2, complete the square then use a trigonometric substitution.
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  6. #6
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    Quote Originally Posted by elven06 View Post
    2.
    \int{\frac{1}{\sqrt{t^2 - 8t + 41}}\,dt} = \int{\frac{1}{\sqrt{t^2 - 8t + (-4)^2 - (-4)^2 + 41}}\,dt}

     = \int{\frac{1}{\sqrt{(t - 4)^2 + 25}}\,dt}.


    Now you actually use a HYPERBOLIC substitution.

    Let t - 4 = 5\sinh{x} so that dt = 5\cosh{x}\,dx.

    Also note that x = \sinh^{-1}{\frac{t - 4}{5}}.


    So the integral becomes

    \int{\frac{1}{\sqrt{(5\sinh{x})^2 + 25}}\,5\cosh{x}\,dx}

     = \int{\frac{5\cosh{x}}{\sqrt{25\sinh^2{x} + 25}}\,dx}

     = \int{\frac{5\cosh{x}}{\sqrt{25(\sinh^2{x} + 1)}}\,dx}

     = \int{\frac{5\cosh{x}}{5\sqrt{\cosh^2{x}}}\,dx}

     = \int{\frac{\cosh{x}}{\cosh{x}}\,dx}

     = \int{1\,dx}

     = x + C

     = \sinh^{-1}{\frac{t - 4}{5}} + C.


    Now convert to its logarithmic equivalent if need be and substitute your terminals.
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