Expert: Paul Klarreich - 7/1/2011

Question
I need to find the definite integral of (4-x)ln((1/2)*x).

I think I use the integration by parts principle. in which case I have:

f(x)= (4-x)     f'(x)=4x-x^2/2
g(x)=?          g'(x)=ln(1/2*x)

Not sure what g(x) would be, and indeed if this is the right method to use.

Any help gratefully received.

Answer
Questioner: Harry
Country: Rotherham, United Kingdom
Category: Calculus
Private: Yes
Subject: Indefinite Integral
Question: I need to find the definite integral of (4-x)ln((1/2)*x).

I think I use the integration by parts principle. in which case I have:

f(x)= (4-x)     f'(x)=4x-x^2/2
g(x)=?          g'(x)=ln(1/2*x)

Not sure what g(x) would be, and indeed if this is the right method to use.

Any help gratefully received.
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Assuming you mean:

(4-x) ln (x/2)

That is

(4-x)(ln x - ln 2) =

(4-x) ln x - (4-x)ln 2 =

4 ln x - x ln x - 4 ln 2 + x ln 2 =

Now you have four terms:

4 ln x: do a standard IBP on this: u = ln x,  dv = dx (disregard the 4)

x ln x: do a standard IBP on this: u = ln x,  dv = x dx

The last two are elementary.  Let me know if you get stuck; or if
this is not the integral you meant.

P.S. You can use THE INTEGRATOR at:

http://integrals.wolfram.com/index.jsp

to check your answer or get hints.