Integral calculus
      The indefinite integral
      Substitution rule
         Evaluating the indefinite integrals using substitution rule examples
         Evaluating the indefinite integrals using substitution rule solutions
Substitution rule
Let f and g are differentiable, with range of g contained in the domain of  f, and let F be antiderivative of  f  such that  F' (u) = f (u) where u = g (x), then 
by applying the chain rule
F(g (x))' = F' (g (x)) g' (x) = f (g (x)) g' (x) .
Thus, obtained is substitution rule
Evaluating indefinite integrals using substitution rule, examples
Evaluate the following indefinite integrals using substitution rule
31.   32.  
33.   34.  
35.   36.  
37.   38.  
39.
 
 
40.
 
Evaluating the indefinite integrals using substitution rule, solutions
Example:   31.  Evaluate
Solution: 
Example:   32.  Evaluate
Solution: 
Example:   33.  Evaluate
Solution:  Let's show that given integral is similar to that solved in the example 31.
 
Example:   34.  Evaluate
Solution: 
Example:   35.  Evaluate
Solution: 
 
Example:   36.  Evaluate
Solution: 
Example:   37.  Evaluate
Solution:  Let's show that given integral is similar to that solved in the above example 36.
Example:   38. Evaluate
Solution: 
Example:  39. Evaluate
 
Solution: 
see the solution of the integral example 23 above.
Example:  40.  Evaluate
 
Solution: 
Calculus contents E
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