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What is the integral of ln(x)? Hint: use parts for this integration

We use parts for this integration even though there is only one term! make ln(x)=1ln(x)

How to do integration by parts

If you have a function that you need to integrate that is two functions of x multiplied by eachother, and youve already tried everything else (inspection and substitution) then you're going to have to use integration by parts. 

Intergrating the function §f(x)g(x) ...

we set one of the fuctions to u' and one to v

normally we'll take the hardest part to integrate as v. 

but in this general example we'll just take f(x)=u' and g(x)=v (putting a ' on a fucntion is to say it has been differentiated) 

The general equation then takes the form 

§(u')v dx = uv - §(v')u dx

or §f(x)g(x) = (§f(x)dx)g(x) - §(§f(x)dx)g'(x)dx which just looks like a mess of symbols.

Sometimes youll have to do this more than once to get the answer, but its a very handy tool in integration! 

So our problem is lnx. To make this by parts we split it up into two terms, 1(lnx) 

Therefore u'=1 and v=lnx so u=x and v'=1/x

This makes the integral....

§-use this as integration sign

§ln(x)dx = xln(x) - §x(1/x)dx

          =xln(x) -§(1)dx

      = xln(x) -x + c 

Joseph W. A Level Physics tutor, A Level Maths tutor, A Level Chemist...

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