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Integral Calculus
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Yes , the function must be continuous everywhere in the domain of integration. Otherwise break the integral. The best example of this is integration of [x] function (Greatest integer function). To be frank , all functions are integrable. The only hitch is that we do not know the methods of integrating it.
Generally , if a smooth graph of the function exisits , it is integrable. You must have heard formulas that are avalable for evaluating definite integrals. They can be applied for any function provided is continous and differentiable in the domain of integration.
If you want to find out indefinite integral of , say f(x) = x / logx , you can pick up any of the formula for evaluating definite integrals. Put the upper limit of integration as "x" and the lower limit of integration as 0. This is beacuse f(0) = 0. Thus you will get the indefinite integral. However , please note that it will not be accurate. As you know , most of these formulas , ask us to break the domain into parts. So , you must evaluate the limit as the number of parts become infinite. By using this method , you will be able to integrate any integrable functions.
Notice , that this is the very basic idea of "integration defined as the summation of terms".
Formulas that you can use : Trapezoidal rule , simpson's 1/3 rule etc.
However , if you want to find whether a function is integrable or not then you must :
1. Check whether the function is continous , differentiable and well-defined (it does not attain abusrd values) in the domain of integration.
If this condition is fulfilled , then the function is integrable , else not.
This again comes from the very basics of integration which relate it to summation of infinite terms.
Cheers,
first know about si ( x ) . it is a new higher function of upper limit x for sinx/x and is denoted by si(x)............
and u know one thing? , to find the antiderivative or indifinite integral of this function, it"s a very futile attempt. it is one of the curious facts of the calculus that some of the basic functions cant be expressed as elementary functions. the function sinx/x belongs to this group. as do cosx/x and e^x / x . and this kind some functions are there... i think u know ...it DOESNT MEAN THAT ANTIDERIVATIVES FOR THIS FUNCTIONS WILL NOT EXIST.(keep this sentence in u r mind fixed).it only means that they cant be expressed in closed form. means in terms of ELEMENTARY FUNCTIONS. this is the reason we r defining the indefinite integral sinx/x as new function by taking limits SI (X) ( MOST BEAUTIFUL FUNCTION).THEN NEXT QUESTION WHAT IS SI(X) FUNCTION ?
ALTHOUGH SI(X) IS NOT AN ELEMENTARY FUNCTION...WE CAN NEVERTHELESS COMPUTE ITS VALUE AND PLOT IT ON THE GRAPH.......
this is done by writing sin function as power series..
x - x^3/3! + x^5/5! ... divide each term by x and integrate it. the required series is si(x).
si(x) = x-x^3/3.3!+x^5/5.5! .......
this is the series which converges for all x. and if we had taken upper limit x . if we increase tihs upper limit with out bound the area of the graph approaches a limit .
si(infinity) = integral 0 to infinity sinx /x is = pie/2
i want to draw these graphs... but dont know how to draw in the system.any way i will give u graph u will definitely understand with perfection.
cheers !!
I'm sorry. sinx/x is integrable , but it's integral cannot be found by conventional means.
See , when you integrate f(x)dx you must make a substitution that fits in the domain of f(x). For example , if you are integrating 1 / sq. rt ( 1 - x^2). You must make a substitution that sees to it , that it does not attain negative value. Consequently , you cannot substitue x as tanx or secx etc. However , you substitute x as sinx or cosx.
Similarly , while integrating F(x) = x^xlnex dx , we make the substitution x^x as k and then take log and then differentiate , thus concluding that it is defined for only positive values of x.
F(x) = sinx/x is a discontinous function at places where sinx = 0. Thus we must make a substitution or transformation that makes it clear that the function will only attain non - zero values , between the domain ( 0 , pi) , (pi , 2 pi) and so on.
However , it is impossible to find out such a substitution or transformation.
Hence , f(x) = sinx/x is not integrable by conventional means.
As a trivial method , expand sinx , divide by x and then integrate , the infinite series. At least , you will get an approximate value of the integral.
hello dear it is non integrable function few other functions are:::: 1. int sinx / x dx 2. int cosx/ x dx 3. int root (sinx) dx 4. int root (cosx) dx 5.int sin (x 2 ) dx 6. int cos (x 2 ) dx 7. int e -x 2 dx 8. int e x 2 dx 10. int root(1+x 3 ) dx 11. int x tan x dx 12. int 1/ log x dx
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even i am also trying for it.but there seems to be very few functions which cannot be integrated.