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Welcome to March Calculus Madness!
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Integration by parts to the rescue!
∫ ln^2 x dx
u = ln^2 x, dv = dx
du = 2 * ln * 1/x dx, v = x
∫ ln^2 x dx
= x * ln^2 x - ∫ 2 * ln x * 1/x * x dx
= x * ln^2 x - ∫ 2 * ln x dx
u = ln x, dv = 2 dx
du = 1/x dx, v = 2 * x
= x * ln^2 x - (2 * x * ln x - ∫ 1/x * 2 * x dx)
= x * ln^2 x - (2 * x * ln x - ∫ 2 dx)
= x * ln^2 x - 2 * x * ln x + ∫ 2 dx
= x * ln^2 x - 2 * x * ln x + 2 * x + C
∫ ln^3 x dx
u = ln^3 x dx, dv = dx
du = 3 * ln^2 x dx, v = x
= x * ln^3 x - ∫3 * ln^2 x * 1/x * x dx
= x * ln^3 x - 3 * ∫ ln^2 x dx
= x * ln^3 x - 3 * (x * ln^2 x - 2 * x * ln x + 2 * x + C)
(see the above)
= x * ln^3 x - 3 * x * ln^2 x + 6 * x * ln x - 6 * x + D
(where D = 3 * C, C and D are constants)
Eddie
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