Integration by Parts: Assigning u & dv with LIPATE Rule

[zenity]

Hi,

I'm a bit confused as to what I should assign u and dv in this integration by parts:

ln(1+x^2)dx

I remember a general rule called the "LIPATE" rule... which is basically Logarithms, inverse trigs, poly, algebra, trig, then exponentials...

Now... would I assign u = ln(1+x^2)? and dv = dx?

That kind of defeats the purpose of integration by parts... doesn't it?

I'm probably not assigning u and dv properly... so if anybody can lend me a hand, thank you!

 

[quantumdude]

Now... would I assign u = ln(1+x^2)? and dv = dx?

Yes.

That kind of defeats the purpose of integration by parts... doesn't it?

No, it works like a charm. Try it and see.

 

[zenity]

Ok... so this is what happens after I work it out

u=ln(1+x)^2 du=2x/(x^2+1)

dv=dx v=x

ln(1+x^2)x - integral [ 2x^2/(1+x^2)

Using a calc, I found out that the integral works out to be x - arctan(x)

... but I don't know how to do it ... any pointers?

 

[quantumdude]

Your "new" integrand is an improper rational function. They should have beat in into your head during your precalc courses that improper rational functions should be decomposed into a polynomial plus a proper rational function, via long division.

Try that, and then see if it's not clear what to do.

 

[zenity]

I'll need a quick refresher on that... I don't remember exactly how to do that...
I'll take a look around on the net. In the meantime, I am 100% stuck on this question:

integral of x^7/(1+x^4)^(3/2)

I've tried assigning u=x^7 and dv="the bottom part"

unfortunately... I don't know where I'm headed.

I'm sorry for all these questions. I have a hard time trying to determine what I should assign to u and dv. Is there a good tutorial online or something that can help me in this area?

 

[quantumdude]

In the meantime, I am 100% stuck on this question:
integral of x^7/(1+x^4)^(3/2)

Don't get so caught up in integration by parts that you forget the other stuff you used to know. This can be done with a simple u-substitution. I'll let you figure out what u is supposed to be.

 

[zenity]

Thanks... but this section in the book insists that we use integration by parts in order to solve these questions... (Although using substitution as a secondary method would be acceptable I think).

I've solved the questions... but I'm now stuck on integration and trigonometric identities...

Integral of sec(x)^4 / tan(x)^2

I've tried using sec(x)^2 = tan(x)^2 + 1 and vice versa, but I don't seem to be getting anywhere with that. If someone could possibly lend a few hinters! Thanks!

 

[VietDao29]

You can just change everything to sin(x) and cos(x), then go from there.
So:
$$\int \frac{\sec ^ 4 x}{\tan ^ 2 x} dx = \int \frac{\cos ^ 2 x}{\sin ^ 2 x \cos ^ 4 x}dx = \int \frac{1}{\sin ^ 2 x \cos ^ 2 x}dx$$
There's sin(x) and cos(x) in the denominator, so you may want to change the numerator a bit so that it also has sin(x) and cos(x).
$$... = \int \frac{\cos ^ 2 x + \sin ^ 2 x}{\sin ^ 2 x \cos ^ 2 x}dx = ...$$
Can you go from here?
--------------------
Or if you want to continue doing it your way, then you can change everything into tan(x).
So:
$$\int \frac{\sec ^ 4 x}{\tan ^ 2 x} dx = \int \frac{(1 + \tan ^ 2 x) ^ 2}{\tan ^ 2 x} dx = \int \frac{dx}{\tan ^ 2 x} + \int \tan ^ 2 x dx + \int 2 dx$$
Now you can change tan(x) to sin(x) and cos(x): tan(x) = sin(x) / cos(x). And use sin²x + cos²x = 1 to solve the problem.
---------------
Anyway, it's always good that you start another thread for another problem, instead of using the old one...
Viet Dao,