- Books Name
- Mathmatics Book Based on NCERT
- KRISHNA PUBLICATIONS
- CBSE Class 12
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
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.