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March 25, 2017

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Integrate the following:

a. Integral from 0 to pi (sin^2)(3x)dx

b. Integral of (x^2)/((x^2 - 4)^3/2)

  • Calc 2 - ,

    a) (sin^2)(3x)
    using cos 2A = 1 - 2sin^2 A
    cos 6x = 1 - 2sin^2 (3x)
    sin^2 3x = 1/2 - (1/2) cos 6x
    so integral sin^2 3x = (1/2)x - (1/12)sin(6x)

    Take it from there, I will let you do the substitution and evaluation

  • Calc 2 - ,

    tried integration by parts on the 2nd, but it got messy, probably made an error
    Gave up and used Wolfram with results of
    log(2(√(x^2-4) + x)) - x/√(x^2-4) + C

    http://integrals.wolfram.com/index.jsp?expr=%28x%5E2%29%2F%28%28x%5E2+-+4%29%5E%283%2F2%29%29&random=false

    tested it and got the original back after differentiating it

    http://www.wolframalpha.com/input/?i=derivative+log%282%28√%28x%5E2-4%29+%2B+x%29%29+-+x%2F√%28x%5E2-4%29

  • Calc 2 - ,

    I tried change of variables (trig functions) on number 2, and got messy also, gave up.

  • Calc 2 - ,

    ∫(x^2)/((x^2 - 4)^3/2 dx
    x = 2coshθ
    dx = 2sinhθ dθ
    x^2-4 = 4sinh^2θ

    ∫4cosh^2θ/8sinh^3θ 2sinhθ dθ

    Keeping in mind that
    arccoshθ = log(z+√(z^2-1))

    I think wolfram's answer is less mysterious.

    Just as trig substitutions are your friend, so are the hyperbolic functions.

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