PPT of Improper Integrals IMPROPER INTEGRAL
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A brief explanation of Improper integrals with appropriate figures. It can also submitted to professor at the time of submission.
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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