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BdMO Online Forum • View topic - Integrability

Integrability

For college and university level advanced Mathematics

Integrability

(1)A function is not differentiable at a point if it is not continuous at that point or it is a corner point(that is L.H.S derivative and R.H.S derivative differ in value).

What are all the cases in which a function doesn't have (a)an indefinite integral or (b)a definite integral?

(2) How do we evaluate the integral $\int x^{-1} dx$ (talking about indefinite integral, a general formula)?
Rabeeb

Posts: 25
Joined: Tue Dec 14, 2010 7:52 pm

Re: Integrability

1.If a function doesn't "have" indefinite integral , then it doesn't "have" definite integral too because they are both same thing except the constant gets cancelled in definite integral and we get a definite value for 2 given limits . Now some functions (like $\ln(\ln x),x^x,\sin(\sin x)$ ) can not be integrated . These are called non-elementary functions. As far as I know , although they can't be integrated , their integral value between 2 limits are calculated approximately . I dont know how though(haven't studied yet! ).
Integral's geometric outcome is calculating area covered by $f(x)$ with $X$ axis . Even we can't calculate non-elementary function's integral directly , that doesn't mean the function doesn't cover any area . We calculate this approximately .

2. While starting derivative chapters you may learn that $\displaystyle \frac{d}{dx}(\ln x)=\frac{1}{x}$ ,
as Integration is the reverse process of differentiation [I think here you will get a nice explanation about reverse relation] ,
$\displaystyle \int \frac{1}{x}dx=\ln x+c$ .
Last edited by Phlembac Adib Hasan on Fri Feb 06, 2015 9:36 am, edited 1 time in total.
Reason: Use \ln and \sin for a nicer look ;)
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photon

Posts: 186
Joined: Sat Feb 05, 2011 3:39 pm
Location: dhaka

Re: Integrability

Well.. But can we integrate all discontinuous functions as well??
Rabeeb

Posts: 25
Joined: Tue Dec 14, 2010 7:52 pm

Re: Integrability

Well, some non-elementary functions can be integrated (Yeah, the blessings of power series. But this is not a general approach, as they don't always converge to the original function.) See more here:
http://en.wikipedia.org/wiki/Nonelementary_integral
http://math.stackexchange.com/questions ... -functions
http://math.stackexchange.com/questions ... integrable
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