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.

This is written in it's simplest form. Do not worry if you forget about the constant in your C4 exam. Most edexcel mark schemes would still give you full marks for this.                                   

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