Thread: Doubt about integration by parts

Hi, colleagues!

On calculus books I read the following formula about the solution of definite integrals by parts:

http://img83.imageshack.us/my.php?im...9234525cz4.jpg

where f, g : [a,b] c R -> R , derivatives of F(x), G(x), are continuous in [a,b].



So, why I can't solve

with g(x) = sin(x), F(x) = 1/cos(x), a= pi/4, b= pi/3

even if f,g are continuous in [pi/4, pi/3] ?

if you look at how to solve an integral of the form f' (x)/f(x) you should see that integration by parts is not required.

Yes, you're right!

I know how to solve it..but I'd like to know why integration by parts fails.. :wink:

- Aforesaid integration by parts permits to solve



even when f,g are not continuous in [a, b] (for example, a=0, b=pi);

the result is identity 0=0.


- Aforesaid integration by parts doesn't permit to solve

:

the result is absurdity 0= -1 !


- Integration by parts permits to solve

, where g(x) = arcsin(x), F(x) = 1,

even if G(x) isn't continuous (in x= +1 and -1).



Why these differences?

Originally Posted by doctor_cat

Hi, colleagues!

On calculus books I read the following formula about the solution of definite integrals by parts:

http://img83.imageshack.us/my.php?im...9234525cz4.jpg

where f, g : [a,b] c R -> R , derivatives of F(x), G(x), are continuous in [a,b].



So, why I can't solve

with g(x) = sin(x), F(x) = 1/cos(x), a= pi/4, b= pi/3

even if f,g are continuous in [pi/4, pi/3] ?

It is not very clear what you are trying to do. If you really are trying to solve for t with
g(x) = sin(x) then it is ia painfully clear that t=1/cos(x) is a solution. If you are trying to integrate then organicgod's observation that you should be able to integrate f'(x)/f(x) without using integration by parts should solve your problem. Integration by parts is just one in an arsenal of techniques for performing integrations. Not all techniques are useful for a given problem.

Yes, I know that..I know how to solve it without integration by parts...

I'd like to know when integration by parts is applicable and why it fails in the foresaid case.

integration by parts is when you try to integrate f(x)g(x). and that one function is not the derivitate of the other.
In this case integration can be used if one of the functions will simplify until you end up with a function that can be integrated.
In your case an integration by parts does not simplify the integral to a form which can be integrated.

so if you had the integral of (x^2) sin x
After 2 succesive integration by parts the x^2 term will simplify to a constant producing a function that can be integrated.

I think you can also use integration by parts for recurring formulas.
which are usually along the lines of In = In+k

so for example In = integral (x^n)*(e^(x^2))
and then performing an integration by parts will lead to a recursive formula which can be used to solve the integral for any n.

Step by step..

Let's suppose we want to use only integration by parts as integration technique, until we obtain a function whose integral is well-known.

Tell me:

can f(x)*g(x) be always integrated using integration by parts or are there conditions to be satisfied about f(x), g(x) and their derivatives ?

I read derivatives must be continuous within the integration range..

am I wrong?