Calc

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Use integration by parts to evaluate the integral.

∫ from 1 to 4 of (√x) (ln√x) dx

Im not really sure what to choose for u and dv.

Thank you so much for the help!!

  • Calc -

    Try to set u to something that gets simpler in the derivative.

    u = ln √x
    du = 1/√x * 1/2√x dx = dx/2x

    dv = √x
    v = 2/3 x3/2

    ∫ u dv = uv - ∫v du
    = 2/3 x3/2 ln √x - ∫ 1/3 √x dx

    = 2/3 x3/2 ln √x - 2/9 x3/2

    = 2/9 x3/2 (3 ln √x - 1)

    F(4) = 16/9 (ln 8 - 1)
    F(1) = -2/9

    F(4) - F(1) = 16/9 ln 8 - 14/9

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