Calculus-Sec-7-1-Solution


 7.1   Integration by Parts                             by SGLee - HSKim, JYLee

                                                                                                              http://youtu.be/jKCAGJ4HqvQ

 

1. 

     http://matrix.skku.ac.kr/cal-lab/sage-grapher-integral.html

 

 

       .

      

                                 .

                                    

                                                            

 2.  

             http://matrix.skku.ac.kr/cal-lab/cal-7-1-2.html

 

 

    .




3. 

 

 

    




4. 

    http://matrix.skku.ac.kr/cal-lab/cal-7-1-4.html 

 

 

     .

      .




5. 

 

 

    

                         .




 6. 

 




7. 

 

  

    

                       .




 8. 

            http://matrix.skku.ac.kr/cal-lab/cal-7-1-21.html

 

 

    

     .




 9.

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

 

      ,

     

                      .




10. 

 

      .

       .




11.  

 

 

    .




12. 

 

  

    

                        .




13. 

      http://matrix.skku.ac.kr/cal-lab/cal-7-1-13.html 

 

  

    

           .




14. 

 

 

     

      




15. 

 

 

    .

      

     by Example 1 Section 7.3.




16. 

 

 

     .

       -①

     Since 

                        ,

      Therefore, from ①, 

                                        .




 17. 

             http://matrix.skku.ac.kr/cal-lab/cal-7-1-19.html

 

     

    .




 18. 

             http://matrix.skku.ac.kr/cal-lab/cal-7-1-20.html

 

    .




 19. 

             http://matrix.skku.ac.kr/cal-lab/cal-7-1-24.html

 

     .




 20. 

              http://matrix.skku.ac.kr/cal-lab/cal-7-1-26.html

 




 21. 

              http://matrix.skku.ac.kr/cal-lab/cal-7-1-22.html

 

 

    .




 22. 

              http://matrix.skku.ac.kr/cal-lab/cal-7-1-25.html

 

    .




 23. 

              http://matrix.skku.ac.kr/cal-lab/cal-7-1-23.html

 

 

    .




24. 

 

       

                             .




25. (a) Establish the reduction formula. 

        .

     (b) Use part (a) to evaluate .

     (c) Use parts (a) and (b) to evaluate .

 

 

    (a) 
         .

         

                          

                          

                          

                          .

          Therefore, 

      (b)

        .

      (c)

        

                         

                         .







26. (a) Establish the reduction formula.

        .

     (b) Find the integral explicitly for .

 

 

     .

     (a)  

                    .

     (b) .




27. (a) Establish the reduction formula

        .

      (b) Find the integral for .

 

     (a)

      

      

                       

                       .

      

                                    .

      .

      .

    (b)

    

                     

                      .




                                               

                                                            <Renyi Institute, Budapest, Hungary, June 2001>

                                                         http://matrix.skku.ac.kr/sglee/album/hungary.html 

 

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