Solution to Problem 11.8-2


© 2001 by Karl Hahn 		KCT logo

2)The problem is to integrate 


   






 dx

Step 1: Multiply numerator and denominator (inside the radical) by  1 - x.  Then simplify so that you don't have the radical over the whole fraction (hint: your numerator will be a perfect square -- take advantage of it). Do all that and then click here to continue.

You should have 


   






 dx  =



  


  1 - x
  ______ dx
 √1 - x2

Step 2: Break it into the sum of two integrals. This is an easy step. Do it and then click here to continue.

You should have 


   






 dx  =



  


    dx
  ______  -
 √1 - x2


  


   x dx
  ______
 √1 - x2

Step 3: Apply trig substitution to the left-hand summand and simple substitution to the right-hand summand. This should be no sweat because you've seen both of these integrals before. Set up the substitutions and then click here to continue.

Your substitution setup for the left-hand summand should be 


   sin(u)  =  x
and 

cos(u) du  =  dx

Your substitution for the right-hand summand should be 

   s  =  1 - x2
and 

  1
-   ds  =  x dx
  2

When you put those substitutions into the integrals you get 


   






 dx  =



  


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


  1
   
  2


 


 ds
  _
 √s

Step 4: Use a trig identity to simplify the left-hand integral. Again we've done this a number of times before. It should be easy. When you've done it click here to continue.

You should have used 

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

to end up with 


   






 dx  =



  



 du  +



  1
   
  2


 


 ds
  _
 √s

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

You should have 


   






 dx  =



         _
  u  +  √s  +  C

Step 6: Substitute back. You have two different variables to substitute back for, u and s. (If you don't remember your substitutions, then click here) When you've done the back-substitution, click here to see the final answer.

When you back substitute  sin(u) = x,  the bare u becomes  arcsin(x).  So the final answer is 


   






 dx  =



                 ______
  arcsin(x)  +  √1 - x2  +  C

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