PPT of Improper Integrals IMPROPER INTEGRAL
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The document discusses improper integrals, specifically focusing on type I with infinite limits of integration and type II with discontinuous integrands. It presents definitions and examples for both types, clarifying how integrals can be classified as convergent or divergent depending on the existence of limits. Additionally, it outlines various scenarios and provides specific examples of improper integrals.
PPT of Improper Integrals IMPROPER INTEGRAL
- 2. Example IMPROPER INTEGRALS Improper Integral TYPE-I: Infinite Limits of Integration TYPE-I: Infinite Limits of Integration ∫ ∞ 1 2 1 dx x Example ∫− 1 1 2 1 dx x TYPE-II: Discontinuous Integrand Integrands with Vertical Asymptotes TYPE-II: Discontinuous Integrand Integrands with Vertical Asymptotes
- 3. IMPROPER INTEGRALS DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 1 = ∫∫ ∞→ ∞ b aba dxxfdxxf )(lim)( Example ∫ ∞ 1 2 1 dx x
- 4. IMPROPER INTEGRALS DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 1 = ∫∫ −∞→∞− b aa b dxxfdxxf )(lim)( Example ∫∞− 0 dxxex
- 5. IMPROPER INTEGRALS DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 1 = ∫∫ −∞→∞− b aa b dxxfdxxf )(lim)( = ∫∫ ∞→ ∞ b aba dxxfdxxf )(lim)( The improper integrals are called convergent if the corresponding limit exists and divergent if the limit does not exist. ∫ ∞ a dxxf )( ∫∞− a dxxf )(
- 6. IMPROPER INTEGRALS DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 1 = ∫∫ ∞→ ∞ t ata dxxfdxxf )(lim)( Example ∫ ∞ 1 1 dx x
- 8. IMPROPER INTEGRALS DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 1 ∫ ∞ ∞− dxxf )( If both improper integrals are convergent ∫ ∞ a dxxf )( ∫∞− a dxxf )( convergent ∫ ∞ ∞− dxxf )( ∫∫ ∞ ∞− += a a dxxfdxxf )()( Example ∫ ∞ ∞− + dx x 1 1 2
- 10. Example IMPROPER INTEGRALS Improper Integral TYPE-I: Infinite Limits of Integration TYPE-I: Infinite Limits of Integration ∫ ∞ 1 2 1 dx x Example ∫− 1 1 2 1 dx x TYPE-II: Discontinuous Integrand Integrands with Vertical Asymptotes TYPE-II: Discontinuous Integrand Integrands with Vertical Asymptotes
- 11. IMPROPER INTEGRALS DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 2 Example ∫ − 5 2 2 1 dx x
- 12. IMPROPER INTEGRALS DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 2 Example ∫ 2/ 0 sec π dxx Example ∫ − 1 0 1 1 dx x
- 13. IMPROPER INTEGRALS DEFINITION OF AN IMPROPER INTEGRAL OF TYPE 2 Example ∫ − 3 0 1 1 dxx