Integrals Class 12 Notes Mathematics Chapter 7

Chapter at a Glance

1. Integration – The process of finding the function f (x) whose differential coefficient w.r.t. ‘x’, denoted by f (x) is given, is called the integration of f(x) w.r.t. x and is written as

Thus, integration is an inverse process of differentiation hence integration is anti of differentiation

Integration by using standard formulae –

2. Integration by Substitution

Integrals of some particular functions

We then equate the coefficient of x & constant terms to find A & B and hence the integral is reduced to one of the known forms

3. Partial Integration

The rational functions which we shall consider here for integration purposes will be those whose denominators can be factorised into linear and quadratic factors.

The following table indicates the types of simpler partial fractions that are to be associated with various kind of rational functions.

4. Integration by parts

i.e., the integral of the product of two functions = (first function) × (Integral of the second function – Integral of {(differential of first function) × (Integral of second function)}This formula is called integration by parts


(ii) If both the functions are trigonometrical, then take that function as the second whose integral is simpler.

(iii) If both the functions are algebraic then take that function as first whose differentiation is simpler.

5. Important Result

6. Definite Integral– The definite integral of f (x) between the limits a to b i.e.

7. Definite Integral as the limit of a sum.

8. General Properties of Definite Integrals –

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