7.8 Improper Integrals

1. .

        http://matrix.skku.ac.kr/cal-lab/cal-7-8-1.html

ValueError: Integral is divergent


2. .

        http://matrix.skku.ac.kr/cal-lab/cal-7-8-2.html

1


3. .

        http://matrix.skku.ac.kr/cal-lab/cal-7-8-3.html

pi 


4. .

        http://matrix.skku.ac.kr/cal-lab/cal-7-8-4.html

ValueError: Integral is divergent.


5. .

        http://matrix.skku.ac.kr/cal-lab/cal-7-8-5.html

sin(1) - cos(1)


6. .

        http://matrix.skku.ac.kr/cal-lab/cal-7-8-6.html

1/log(2)1/log(2) 


7. .

        http://matrix.skku.ac.kr/cal-lab/cal-7-8-7.html

1/2 


8. .

        http://matrix.skku.ac.kr/cal-lab/cal-7-8-8.html

2


9. .

        http://matrix.skku.ac.kr/cal-lab/cal-7-8-9.html

-gamma(n, x)


 10. .

         http://matrix.skku.ac.kr/cal-lab/cal-7-8-10.html

-1/4


11. .

         http://matrix.skku.ac.kr/cal-lab/cal-7-8-11.html

0


12. .

         http://matrix.skku.ac.kr/cal-lab/cal-7-8-12.html

pi


13. .

         http://matrix.skku.ac.kr/cal-lab/cal-7-8-13.html


1


14. .

         http://matrix.skku.ac.kr/cal-lab/cal-7-8-14.html

1/2*pi


15. .

  

  

  convergent


16. .

         http://matrix.skku.ac.kr/cal-lab/cal-7-8-16.html

ValueError: Integral is divergent


17. .

         http://matrix.skku.ac.kr/cal-lab/cal-7-8-17.html

1/2*log(2)


 18. .

         http://matrix.skku.ac.kr/cal-lab/cal-7-8-18.html

32/3


19. .

 =


20. .

  .

  convergent


21.

         http://matrix.skku.ac.kr/cal-lab/cal-7-8-21.html

1/2*pi1/2*pi 


 22. .

         http://matrix.skku.ac.kr/cal-lab/cal-7-8-22.html

log(2) 


23. .

          http://matrix.skku.ac.kr/cal-lab/cal-7-8-23.html

2/3


24. .

         http://matrix.skku.ac.kr/cal-lab/cal-7-8-24.html

ValueError: Integral is divergent.


25. .

         http://matrix.skku.ac.kr/cal-lab/cal-7-8-25.html

-1/2


26. .

         http://matrix.skku.ac.kr/cal-lab/cal-7-8-26.html

5/2*pi


27. .

         http://matrix.skku.ac.kr/cal-lab/cal-7-8-27.html

  (: positive integer) =.

-1/2*x^(2*n)*gamma(n, x^2)/(x^2)^n


28. .

     http://matrix.skku.ac.kr/cal-lab/cal-7-8-28.html

ValueError: Integral is divergent


29. .

    http://matrix.skku.ac.kr/cal-lab/cal-7-8-29.html

 .

1/23


 30. .

         http://matrix.skku.ac.kr/cal-lab/cal-7-8-30.html

8/15 


31. .

    http://matrix.skku.ac.kr/cal-lab/cal-7-8-31.html

 

      

      Therefore,.

ValueError: Integral is divergent.


32. .

    http://matrix.skku.ac.kr/cal-lab/cal-7-8-32.html

 .

2


33. .

         http://matrix.skku.ac.kr/cal-lab/cal-7-8-33.html

sqrt(pi)


34. Show that .

Proof)

      

       Therefore, .


35. Prove that   converges. (Hint: let

  

       

                   

       

       

       

       

       


36. .

    http://matrix.skku.ac.kr/cal-lab/cal-7-8-36.html

1/2*a^2*arcsin(x/sqrt(a^2)) + 1/2*sqrt(a^2 - x^2)*x1/2*a^2*arcsin(x/sqrt(a^2)) + 1/2*sqrt(a^2 - x^2)*x