Maths - Integration by Parts

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Question 1
ç e^2x Cos(4x) dx

Question 2)
ç ãx sin(x) dx

Question 3)
ç x^2 sin(x) dx

  • Maths - Integration by Parts -

    I'll do #1, you can show us where you get stuck on the others.

    Let I = ∫e^2x cos4x dx

    u=cos 4x
    du = -4sin4x dx

    dv = e^2x dx
    v = 1/2 e^2x

    I = uv - ∫v du
    = 1/2 e^2x cos4x - ∫-2e^2x sin4x dx

    Now do it again, letting u = sin4x

    I = 1/2 e^2x cos4x + (e^2x sin4x - ∫4e^2x cos4x dx)
    = 1/2 e^2x cos4x + e^2x sin4x - 4I
    5I = 1/2 e^2x cos4x + e^2x sin4x
    I = 1/10 e^2x (cos4x + 2sin4x)

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