Integral calculus
The indefinite integral

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