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Integration by Parts


Date: 05/02/99 at 16:03:55
From: Steve Brown
Subject: Integration by Parts

Recently on an study exam for the AP Calc AB test, I ran across the 
following exercise: the general integral of x*ln(x)-1.  I've 
determined that I need to do it by parts, since substitution fails.


Date: 05/03/99 at 22:08:32
From: Doctor Jeremiah
Subject: Re: Integration by Parts

Hi Steve:

I think you are right. Substitution did not work for me either. So we 
will do it by parts.

The first thing to do is to simplify the integral. We need to break it 
up so that it will be easier to solve.

 /                  /              /
 | x*ln(x) - 1 dx = | x*ln(x)*dx - | dx
 /                  /              /

In order to integrate by parts you need to break the integral into two 
pieces called "u" and "dv" .  For the integral of x*ln(x)*dx we will 
choose u = ln(x) and dv = x dx .

 /                  /          /
 | x*ln(x) - 1 dx = | u * dv - | dx
 /                  /          /

But we will also need to find "du" and "v" which can be found like 
this:

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

When you integrate by parts you follow the rule that says the integral 
of v * du is u*v minus the integral of v*du:

 /                  /
 | u * dv = u * v - | v * du
 /                  /

So we will substitute in u*v minus the integral of v*du where we had 
the integral of v * du:

 /                          /          /
 | x*ln(x)-1 dx = u * v - | v * du - | dx
 /                          /          /

 /                                        /                            
/
 | x*ln(x)-1 dx = ln(x) * (1/2 * x^2)-|(1/2 * x^2) * (1/x * dx)-|dx
 /                                        /                            
/

 /                                              /        /
 | x*ln(x)-1 dx = ln(x) * (1/2 * x^2) - 1/2 * | x dx - | dx
 /                                              /        /

And now we can solve the integrals:

 /
 | x*ln(x)-1 dx = ln(x) * (1/2 * x^2) - 1/2 * (1/2 * x^2) - (x)
 /

Now to test our work.  If we differentiate ln(x) * (1/2 * x^2) - 1/2 * 
(1/2 * x^2) - (x) with respect to x we get:

d/dx( ln(x) * (1/2 * x^2) ) - 1/4 * d/dx(x^2) - d/dx(x)
(1/2 * x^2) * d/dx( ln(x) ) + ln(x) * d/dx( 1/2 * x^2 ) - 1/4 * 2x - 1
(1/2 * x^2) * 1/x + ln(x) * 1/2 * d/dx( x^2 ) - 1/4 * 2x - 1
1/2 * x + ln(x) * x - 1/2 * x - 1
ln(x) * x - 1

Which is what we started with.  Let me know if you need more details 
on how I solved it.

- Doctor Jeremiah, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Calculus

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