calculus

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Evaluate the indefinite integral.

integral of 4e^(4x)sin[e^(4x)]dx

  • calculus -

    let y = 4 x
    dy = 4 dx
    so
    integral of e^y sin y dy

    try e^y cos y
    derivative = -e^y sin y + e^y cos y
    so
    try ae^y cos y + be^y sin y
    derivative = -ae^ysiny+ae^ycosy+be^ycosy+be^ysiny
    we need a=-b to get rid of cos terms
    so
    -b e^y cos y +b e^y sin y
    so
    be^y sin y - be^y cos y
    try derivative
    b e^y cos y+be^y sin y+be^y siny-be^y cosy
    =2 b e^y sin y
    so let b = 1/2 and we are there


    =(1/2)e^y [ sin y-cos y] + c

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