Integration

1.) Find

Solution. We have

Set  . Then we have  which implies

Going back to the inconstant x, we get

2.)

3.)  
   Now recall the trig individuation,
                                 
  
 
4.)  

5.)     
 
6.)
     
7.)

7.)  

8.)  

9.)

10.)

in this form we whoremonger do the inbuilt victimization the substitution .  Doing this gives,
                                                

11.)
   

 
  
12.)    
 
     
13.)    
   
1. Lop turned a se stick outt-squared(x) factor and move it to the right.

2. Convert the remaining secants to tangents with the Pythagorean identity,

3. Solve by substitution, where u = tan(x) and

14.)

Making the substitution u = sin x, du = cos xdx and using the identity , we have got
      
15.)
Using identities  and , we can write:
     
depend the underlyings in the latter expression.
      
To find the full , we make the substitution u = sin 2x, du = 2cos 2xdx. Then
      
Hence, the initial integral is
      
16.) Calculate the integral .
Solution.
We can write:
    
Transform the integrand using the identities
     
We get
      
17.) Evaluate the integral .

Solution.




We delectation the identity  to transform the integral. This yields
      
Calculate the integral .

Solution.
Using the identity , we have
      

18.) Calculate the integral .

Solution.
We use the reduction formula
      
Hence,
      
The integral  is a table integral which is have-to doe with to . (It can be easily found usingthe universal trigonometric substitution .) As a result, the integral becomes
      
18.) Evaluate the integral .

Solution.
We use the reduction formula
      
Hence,
      
20. Compute .

Solution.
      

21.) Compute .

Solution.
Use the identity . Then
      
Since  (see Example 9) and  is a table integral equal to , we obtain the following complete answer:
      

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