Solution to Problem 11.8-1


© 2001 by Karl Hahn 		KCT logo

1) The problem is to integrate 


   


 x arcsin(x) dx

Step 1: Apply integration by parts. The assignment of parts that is useful is 



   u  =  arcsin(x)
and 

          dx
du  =   ______
       √1 - x2



   dv  =  x dx
and 

      1
v  =   
      2



 x2

Set up the integration by parts using these assignments, then click here to continue.

You should have 


   


 x arcsin(x) dx  =


  1
   
  2



 x2 arcsin(x)  -



  1
   
  2


 


  x2 dx
  ______
 √1 - x2

Step 2: Apply trig substitution to the remaining integral. Your substitution is 


   sin(t)  =  x
and 

cos(t) dt  =  dx

Put these substitutions in for x in the remaining integral, the click here to continue.

You should have 


   


 x arcsin(x) dx  =


  1
   
  2



 x2 arcsin(x)  -



  1
   
  2


 


  sin2(u)cos(u)
   ___________  du
  √1 - sin2(u)

Step 3: Use the basic identity that puts cosine in terms of sine to simplify the above. When you're done simplifying, click here to continue.

You should have 


   


 x arcsin(x) dx  =


  1
   
  2



 x2 arcsin(x)  -



  1
   
  2


 



 sin2(u) du

Step 4: Use the identity for sine squared to make this easier to integrate, then click here to continue.

You should have 


   


 x arcsin(x) dx  =


  1
   
  2



 x2 arcsin(x)  -



  1
   
  4


 



 (1 - cos(2u)) du  =



                       1
                        
                       2



 x2 arcsin(x)  -



  1
   
  4


 



 du  +



  1
   
  4


 



 cos(2u) du

Step 5: Integrate. Both the integrals you have now are easy. Integrate them and then click here to continue.

You should have 


   


 x arcsin(x) dx  =


  1
   
  2



 x2 arcsin(x)  -



  u     1
     +    sin(2u)  +  C
  4     8

Step 6: To substitute back you must first apply the double angle formula to  sin(2u). After that you'll have to use another identity to put the  cos(u)  term in the result into an expression in sine. Do all that, then click here to continue.

The double angle formula tells you that 

   sin(2u)  =  2 sin(u)cos(u)
But that still has a cosine in it. Our back-substitution requires everything to be put into terms of  sin(u).  If you then use the identity 

               ___________
   cos(u)  =  √1 - sin2(u)

applying all that to the expression you have for the integral you should get 

   


 x arcsin(x) dx  =


  1
   
  2



 x2 arcsin(x)  -



  u     1
     +   
  4     4


         ___________
 sin(u) √1 - sin2(u)

Step 7: Substitute back. The original substitution was  sin(u) = x.  Make the back substitution, then click here for the final answer.

If your original substitution was  sin(u) = x,  then  sin(u)  gets replaced with x, and u gets replaced with  arcsin(x).  With that your final answer is 


   


 x arcsin(x) dx  =


  1
   
  2



 x2 arcsin(x)  -



  1
    arcsin(x)  +
  4


  1
   
  4


    ______
 x √1 - x2 + C

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