Shankershinh Vaghela Bapu Institute of Tecnology

Calculus

IMPROPER INTEGRALGuidance by – MAULIK PRAJAPATI

Presented by KRISNADITYA RANAYATIN DESAILAKSHMI VIMALKISHAN PATELSAGRIKA MAURYASHREY PATEL

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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

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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

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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

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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

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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

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Vertical P-integral test

1. Converges if P < 1

2. Diverges if P > 1

The integral

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Thanks for your anticipation