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Calculus Integration Solutions: Trig Substitution & Partial Fractions

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9강 과제풀이
[[7.3절 19번]]
풀이)
Evaluate the integral

  
 .




  sec ,  ≤      ≤   



⇒    sec tan  , 
    tan


  
tan

tan 
    sec tan      



sec 
sec 








sin  
 sin         cos      






 

  
     sin cos     sec      ⋅   










  
  sec    




 
 
- 17 -
[[7.3절 31번]]
풀이)
Evaluate the integral
      .
  



      
        
    
    sin ⇒   cos 
       cos
          sin  ⋅cos⋅cos 
  
    sin  sin cos  
  cos   sin cos   sin  cos   
  cos     sin cos     sin  cos  













 cos       cos       sin      sin cos  

  cos ⇒   sin 
 sin cos               cos   




 sin  cos     sin        cos        sin  





     ⋅ sin cos  




     ⋅ sin cos⋅ cos  sin    





     sin cos   sin  cos  




   sin cos  cos    sincos  sin cos 


   sin cos   cos  sincos  sin cos  


   sin cos   cos    cos  sin cos  


   sin cos   cos  



    
    
 
    
 sin          









 sin        
             






 


- 18 -
 

[[7.4절 23번]]
풀이)

 .

    
Evaluate the integral



  

 
.

 
     
 
                          
⇒         






        ≡        

  
    
    




 



 ln     ln     tan     



[[7.8절 19번]]
Determine whether
∞
   is convergent or divergent.
 
∞
∞
풀이)

∞
          
 
∞
 
∞

 


    lim    lim       lim       
 
→∞
∞
∞
  

 
→∞


 
→  ∞
    lim    lim       lim      
 
→∞

Therefore
∞
→∞


 
 
∞
→  ∞
Determine whether


 



 is convergent or divergent.
 


풀이)
∞
 
                
∞
[[7.8절 39번]]
  

 





        
 












  





  lim     lim     lim      ∞
 









Therefore
 → 


 → 

 → 


 is divergent.
 



- 19 -

: convergent
[[7.8절 70번]]
Find the values of  for which the integral
∞

 converges and
 
ln


evaluate the integral for those values of .
풀이)

  ln ⇒     ,    ⇒   


∞


 


 ln 

∞

 

if   

 




diergent if  ≤ 
- 20 -
by Example 4 in page 545.
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