Using the Chain Rule: Define Functions

Date: 24 May 1995 04:29:09 -0400
From: Milli Kutuphane
Subject: Integrating sin(ln(4x+5)) and ln(sinx).

1)if y=sin(ln(4x+5))
   y'=?

2) what is the integral of ln(sinx) between e and 0 ?

Date: 24 May 1995 17:51:20 -0400
From: Dr. Sydney
Subject:Integration using the Chain Rule

Dear Milli,

Hello!  To do the first problem you sent, you have to use the chain rule.
One way to do the chain rule is to explicitly define all of the functions
you are working with.  Let f(x) = sin x; let g(x) = ln x; let h(x) = 4x + 5 

then y=f(g(h(x)))          (*)

In other words, y is f composed with g composed with h. 

Then by the chain rule, 

y' = f'(g(h(x)))*g'(h(x))*h'(x)               (**)

Notation here is a bit funny because I am using the same notation to mean
two different things!  In equation (*), I mean f(g(h(x))) to be f composed
with g composed with h.  You often see this written as f o g o h.  In
equation (**), I mean f'(g(h(x))) to be the derivative of f evaluated at the
point g(h(x)); likewise I mean g'(h(x)) to be the derivative of g evalueated
at the point h(x).  Do you see the difference?

Knowing this, can you finish up the problem?  If you need more help or want
to check your answer, feel free to write back.
 

On to your second question!
I don't see an easy way to integrate this one!  The standard tricks of
integration seem not to work.  One thing to remember is that ln is not
defined at 0, so you are dealing with an improper integral here.  I don't
have any tables of integrals with me, but you could check to see if you could
find this there.  Probably there is no nice solution, though -- that is my
guess.  

Hope this helps!

--Sydney, "dr. math"