10. Evaluation of Definite Integrals by Substitution and more methods

Evaluation of Definite Integrals by Substitution and more methods:

To evaluate  by substitution, the steps could be as follows:

1. Consider the integral without limits and substitute, y = f (x) or x = g(y) to reduce

the given integral to a known form.

2. Integrate the new integrand with respect to the new variable without mentioning

the constant of integration.

3. Re-substitute for the new variable and write the answer in terms of the original

variable.

4. Find the values of answers obtained in (3) at the given limits of integral and find

the difference of the values at the upper and lower limits. Although we will not formally prove this theorem, we justify it with some calculations here. From the substitution rule for indefinite integrals, if F(x) is an anti-derivative of f(x) we have

 and we have the desired result.Using substitution to evaluate a definite integral requires a change to the limits of integration. If we change variables in the integrand, the limits of integration change as well.

Alternatively, first we transform the integral and then evaluate the transformed integral with new limits.