calc

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What is the integral of 20 * (sinx)^3 * (cosx)^2?

  • calc -

    y' = 20 * (sinx)^3 * (cosx)^2
    = 20 sinx(sinx)^2(cosx)^2
    = 20sinx(1 - cos^2 x)cos^2 x
    = 20sinx cos^2 x - 20sinx cos^4 x

    y = -20/3(cos^3 x) + 20/3(cos^5 x) + C, where C is a constant

  • calc -

    let u = sin^2 x
    then du = 2 sin x cos x dx
    let dv = -3 sin x cos^2 x dx
    then v = cos^3 x
    u dv = -3 sin^3 x cos^2 x dx
    so
    let u = -20/3 sin^2 x
    then du = -40/3 sin x cos x dx
    let dv = -(1/3)sin x cos^2 x dx
    then v = cos^3 x
    now by parts
    int u dv = u v - int v du
    int (20 sin^3 x cos^2 x dx) =
    -(20/3)sin^2 x cos^3 x +(40/3)int(cos^4 x sin x dx)
    well remember the first term at the end and the (40/3) and work on the integral
    int cos^4 x sin x dx = -(1/5)cos^5 x

    I think you can get it from there

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