7.1 Integration by Parts

1..

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

 ,

      

                 

                 .

      

      (log(x)^2 - 2*log(x) + 2)*x


2. .

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

 ,

       ,

      ,

      .

      

      2/49*(7*x-1)*e^(7*x) + 1/343*(49*x^2 - 14*x + 2)*e^(7*x)

      + 1/7*e^(7*x)2/49*(7*x - 1)*e^(7*x) + 1/343*(49*x^2 - 14*x + 2)*e^(7*x) + 1/7*e^(7*x)


3. .

  .

      

                       .


4. .

 .

       .


5. .

 .

      .


6. .

 .

      


7. .

 


8. .

 

  


9. .

 

                        

                        .


10. .

  

                        

                        .


11. .

  .


12. .

  

                        

                        .


13. .

   

                     

                     

                     .


14. .

    

                 

                 

                .

      Thus,

           

                      

                      

                      .

      Thus, .

      Thus, .

      Therefore,

                           .


15. .

  

         


16.  .

  

                    .

 

17. .


18. .

 .

      

      

       

      Therefore,

                               .


19. .

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

 .

      

      1/2*x^2*arccos(x) - 1/4*sqrt(-x^2 + 1)*x + 1/4*arcsin(x)


20. .

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

 .

      

      -1/10*(2*sin(2*x) - cos(2*x) + 5)*e^(-x)


21. .

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

 .

      

      x*log(sqrt(x) + 1) - 1/2*x + sqrt(x) - log(sqrt(x) + 1)


22. .

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

 .

      

      1

        

23. .

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

 .

      

      1/4*pi - 1/2*log(2)


 24. .

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

 .

      

      2*pi - 4


25. .

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

 .

      

      4*log(2)^2 - 2*log(2) + 15/32


26. .

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


27. (a) Establish the reduction formula. 

        .

    (b) Use part (a) to evaluate .

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

 (a) .

         

                     

                     

                     

                     .

          Therefore,

      (b) .

      (c)

                    

                    .


28. (a) Establish the reduction formula.

        .

    (b) Find the integral explicitly for .

 .

       

                    .


29. (a) Establish the reduction formula

        .

    (b) Find the integral for .

 

      

                 

                 .

      

                 

                .

      .

      .