A couple of integrals ( I think I might have one correct)

[Opus89]

I have a couple of indefinite integrals that I need a little help with.
The first one is:
the integral of dx/(xlnx). In this problem, i let u= lnx. Thus, my du = (1/x)dx or dx/x. When I subtitute, I get du/u which = ln(u). So shouldn't my answer be ln(lnx) + C?

The second integral is pretty tricky. It is the integral of dx/(x^2 -4x +7). I can't seem to factor the bottom into anything and I haven't found a u substitution that works. My teacher gave a hint and said there is an arctan in it.

Any suggestions for either problem?
Thanks

 

[PAULK]

dx
-------
x ln x

u = ln x, dx/x = du

du
----- = ln u = ln ln x
u

Yep, looks OK to me:
.....................
dx
-----------
x^2 -4x +7

Complete the square on x^2 -4x + 7

x^2 -4x + 7

x^2 -4x + 4 - 4 + 7

(x - 2)^2 + 3

Now let u = x - 2, you get

du
--------
u^2 + 3

and compare that with

du
----------
u^2 + a^2

That should do it.

 

[Opus89]

Thanks so much for the input. I'm a little confused on the completing the square part, I always thought when you completed the square, you would wind up with an answer like x= 2+/- sqrt(3)

 

[PAULK]

Thanks so much for the input. I'm a little confused on the completing the square part, I always thought when you completed the square, you would wind up with an answer like x= 2+/- sqrt(3)

There are many applications of CTS. Solving quadratic equations is only one of them. But it comes up often enough in other situations to merit some practice. (Kind of like working on your backhand volley or something.)