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
 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