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Evaluate the integral of x*ln(1+x) dx by first making a substitution and then using integration by parts.

I let u=1+x and du = dx but then ln(u) du can't integrate to anything?

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1 answer

  1. you need to get rid of that pesky ln(1+x). So,

    u = ln(1+x)
    du = 1/(1+x) dx
    dv = x dx
    v = 1/2 x^2

    ∫u dv = uv - ∫v du
    = 1/2 x^2 ln(1+x) - 1/2 ∫x^2/(1+x) dx

    which is not so hard, eh?

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