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
21.   22.  
23.   24.  
25.   26.  
27.   28.  
29.   30.  
Evaluating the indefinite integrals using substitution rule, solutions
Example:   21.  Evaluate
Solution: 
Example:   22.  Evaluate
Solution: 
Example:   23.  Evaluate
Solution: 
Example:   24.  Evaluate
Solution: 
Example:   25.  Evaluate
Solution: 
Example:   26.  Evaluate
Solution: 
Example:   27.  Evaluate
Solution: 
Example:   28.  Evaluate
Solution: 
Example:   29.  Evaluate
Solution: 
This integral can also be solved using decomposition of rational function into a sum of partial fractions.
 
Example:   30.  Evaluate
Solution: 
Calculus contents E
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