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Integrate the following function by parts and reduce it to it's simplest form. f(x) = ln(x).

First note that ln(x) = 1*ln(x), this is in the form         u*dv/dx.

Let dv/dx = 1 and u = ln(x)                                   du/dx = 1/x from the standard results and v = x by integration.

Substituting into the formula

integral(u*dv/dx)dx = uv - integral(v*du/dx)dx we get

Integral(ln(x))dx = x*ln(x) - integral( x/x )dx                                  = x*ln(x) - integral(1)dx                                        = x*ln(x) - x + C                                                  = x(ln(x) - 1) + C

Ryan B. A Level Maths tutor, A Level Further Mathematics  tutor

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Integrate the following function by parts and reduce it to it's simplest form. f(x) = ln(x).

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