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Shankershinh Vaghela Bapu Institute of Tecnology
Calculus
IMPROPER INTEGRALGuidance by – MAULIK PRAJAPATI
Presented by KRISNADITYA RANAYATIN DESAILAKSHMI VIMALKISHAN PATELSAGRIKA MAURYASHREY PATEL
140750116048140750111001140750111002140750111004140750111003140750111005
INTRODUCTION In most of applications of engineering and science there occurs special functions, like gamma functions ,beta functions etc , which are in the form of integrals which are of special types in which the limits of integration are infinity or the integrand becomes unbounded within the limits . Such type of integrals are known as improper integrals. Convergence of such integrals has an important and main roll rather than divergent integral . we shall discuss about the type of improper integrals
DEFINATION The definite integral is said to be improper integral if one or both limits of integration are infinite and/or if the integrand integral is unbounded on the interval
EXAMPAL
TYPES OF INTEGRALS
1)When upper limit is infinity2)When lower limit is infinity3)When both limits of integration are
infinity4)When integral is Unbounded
1) When upper limit is infinityNow is F is Continuous on an interval [a,….) then an improper integral can be define as follow
If this limit exist , we say that I is Convergent ; if not , it is divergent
EXAMPAL
2) When lower limit is infinityNow is F is Continuous on an interval (..,b] then an improper integral can be define as follow
If this limit exist , we say that I is Convergent ; if not , it is divergent
EXAMPAL
3)When both limits are infinityNow is F is Continuous on an interval (..,b] then an improper integral can be define as follow
If this limit exist , we say that I is Convergent ; if not , it is divergent
EXAMPAL
4) When Integrand is Unbounded
A) If f(x) is continuous on [ a , b)
If limit exist Integral is Converges , otherwise it is diverges.
B) If f(x) is continuous on ( a , b]
If limit exist Integral is Converges , otherwise it is diverges.
C) If f(x) is continuous on [ a , b] and not bounded at the point C E ( a ,b) then we can write
If limit exist Integral is Converges , otherwise it is diverges.
Horizontal P-integral test
The integral
1. Converges if P > 1
2. Diverges if P < 1
_
Vertical P-integral test
1. Converges if P < 1
2. Diverges if P > 1
The integral
_
Thanks for your anticipation