integration by substitution

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startingover

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integral (1/xlnx) dx

Let u = lnx
du = 1/x

1/x integral 1/u dx
1/x * ln dx
1/x * lnx + C

These are really confusing me...
 
Let \(\displaystyle u=ln(x), \;\ x=e^{u}, \;\ dx=e^{u}du\)

Then you have:

\(\displaystyle \L\\\int\frac{1}{e^{u}\cdot{u}}e^{u}du\)

Now, can you finish?.
 
Differentiate \(\displaystyle \L \frac{{\left( {\ln (x)} \right)^2 }}{2}\).

What did you get for an answer? How does that relate to your problem?
 
startingover said:
integral (1/xlnx) dx
Click to expand...
You simply must get parentheses in the right places. (1/x)ln(x) would have been more clear.

Let u = lnx
du = 1/x
Click to expand...
Careless. du = (1/x)dx It isn't just window dressing.
 
Hello, startingover!

\(\displaystyle \L\int\frac{1}{x}\)\(\displaystyle (\ln x)\,dx\)
Click to expand...

We have: \(\displaystyle \L\:\int (\ln x)\left(\frac{dx}{x}\right)\)

Let: \(\displaystyle \,u\,=\,\ln x\;\;\Rightarrow\;\;du\,=\,\frac{dx}{x}\)


Substitute: \(\displaystyle \L\int\underbrace{(\ln x)} \underbrace{\left(\frac{dx}{x}\right)}\)
. . . . . . . . . .\(\displaystyle \L\int\;u\;\;du\;\;=\;\;\frac{1}{2}u^2\,+\,C\)

Back-substitute: \(\displaystyle \L\:\frac{1}{2}\left(\ln x\right)^2\,+\,C\)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

If the problem is: \(\displaystyle \L\:\int\frac{dx}{x\cdot\ln x}\)

then we have: \(\displaystyle \L\:\int\frac{1}{\ln x}\cdot\frac{dx}{x}\)

Let: \(\displaystyle u\,=\,\ln x\;\;\Rightarrow\;\;du \,=\,\frac{dx}{x}\)


Substitute: \(\displaystyle \L\:\int\underbrace{\frac{1}{\ln x}}\cdot\underbrace{\frac{dx}{x}}\)

. . . . . . . . . . \(\displaystyle \L\int\frac{1}{u}\,du\;=\;\int\frac{du}{u}\;=\;\ln u\,+\,C\)

Back-substitute: \(\displaystyle \L\:\ln(\ln x)\,+\,C\)

 
The problem is actually written as:
1 over xlnx dx

So I tried my best to show it that way with the parenthesis.

integral (1)/(xlnx) dx
 
Good call. Please accept my apologies for chastizing your correct notation. I just didn't believe it.

Unfortunately, often it is insufficient to type so that anyone can understand. It must be so that no one can misunderstand. I realize that is a very high standard.

A little LaTeX would have cleared that up nicely.

I wrote an email the other day trying to clear up a misconception. Someone thought a rate was $100. I tried to convince them it was $10. How could ANYONE know that either $100 or $10 was or wasn't just a typo? I finally resorted to overt redundancy - in numbers and in spelling. They understood my message, but someone thought I was being demeaning or insulting. Sometimes you can't win.

Again, sorry for misreading your properly written problem statemt. Learn some LaTeX. Fractions are easy.
 
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