# NCERT Class 12-Mathematics: Exemplar Chapter –7 Integrals Part 1

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## 7.1 Overview

**7.1.1**

Let The, we write. These integrals are called indefinite integrals or general integrals, C is called a constant of integration. All these integrals differ by a constant.

**7.1.2** If two functions differ by a constant, they have the same derivative.

**7.1.3** Geometrically, the statement represents a family of curves. The different values of correspond to different members of this family and these members can be obtained by shifting any one of the curves parallel to itself. Further, the tangents to the curves at the points of intersection of a line with the curves are parallel.

**7.1.4 Some properties of indefinite integrals**

(i) The process of differentiation and integration are inverse of each other, , where C is any arbitrary constant.

(ii) Two indefinite integrals with the same derivative lead to the same family of curves and so they are equivalent. So if and *g* are two functions such that then and are equivalent.

(iii) The integral of the sum of two functions equals the sum of the integrals of the functions i.e.,

(iv) A constant factor may be written either before or after the integral sign, i.e.,

is a constant.

(v) Properties (iii) and (iv) can be generalised to a finite number of functions and the real numbers, giving

**7.1.5 Methods of integration**

There are some methods or techniques for finding the integral where we can-not directly select the ant derivative of function by reducing them into standard forms. Some of these methods are based on

1. Integration by substitution

2. Integration using partial fractions

3. Integration by parts.

**7.1.6 Definite integral**

The definite integral is denoted by , where *a* is the lower limit of the integral and is the upper limit of the integral. The definite integral is evaluated in the following two ways:

(i) The definite integral as the limit of the sum

(ii) , if is an ant derivative of .

**7.1.7 The definite integral as the limit of the sum**

The definite integral is the area bounded by the curve, the ordinates and the *x*-axis and given by

Where

**7.1.8 Fundamental Theorem of Calculus**

(i) Area function: The function denotes the area function and is given by

(ii) First Fundamental Theorem of integral Calculus

Let *f* be a continuous function on the closed interval and let be the area function . Then .

(iii) Second Fundamental Theorem of Integral Calculus

Let be continuous function defined on the closed interval and be an ant derivative of

**7.1.9 Some properties of Definite Integrals**

(ii)