**CBSE Class 12 Commerce Mathematics Integrals**

**CBSE Class 12 Commerce Mathematics Integrals:-** we provide complete details of CBSE Class 12 Commerce Mathematics Integrals in this article.

**CBSE Class 12 Commerce Mathematics Integrals**

CBSE Class 12 Commerce Mathematics IntegralsDifferential Calculus is centred on the concept of the

derivative. The original motivation for the derivative was

the problem of defining tangent lines to the graphs of

functions and calculating the slope of such lines. Integral

Calculus is motivated by the problem of defining and

calculating the area of the region bounded by the graph of

the functions.

If a function f is differentiable in an interval I, i.e., its

derivative f ′exists at each point of I, then a natural question

arises that given f ′at each point of I, can we determine

the function? The functions that could possibly have given

function as a derivative are called anti derivatives (or

primitive) of the function. Further, the formula that gives

all these anti derivatives is called the indefinite integral of the function and such

process of finding anti derivatives is called integration. Such type of problems arise in

many practical situations. For instance, if we know the instantaneous velocity of an

object at any instant, then there arises a natural question, i.e., can we determine the

position of the object at any instant? There are several such practical and theoretical

situations where the process of integration is involved. The development of integral

calculus arises out of the efforts of solving the problems of the following types:

(a) the problem of finding a function whenever its derivative is given,

(b) the problem of finding the area bounded by the graph of a function under certain

conditions.

These two problems lead to the two forms of the integrals, e.g., indefinite and

definite integrals, which together constitute the Integral Calculus.

There is a connection, known as the Fundamental Theorem of Calculus, between

indefinite integral and definite integral which makes the definite integral as a practical

tool for science and engineering. The definite integral is also used to solve many interesting

problems from various disciplines like economics, finance and probability.

### **CBSE Class 12 Commerce Mathematics Integrals:-****Integration as an Inverse Process of Differentiation**

Integration is the inverse process of differentiation. Instead of differentiating a function,

we are given the derivative of a function and asked to find its primitive, i.e., the original

function. Such a process is called integration or anti differentiation.

### **CBSE Class 12 Commerce Mathematics Integrals:-****Geometrical interpretation of indefinite integral**

Let f (x) = 2x. Then ∫ f (x) dx = x2 + C . For different values of C, we get different

integrals. But these integrals are very similar geometrically.

Thus, y = x2 + C, where C is arbitrary constant, represents a family of integrals. By

assigning different values to C, we get different members of the family. These together

constitute the indefinite integral. In this case, each integral represents a parabola with

its axis along y-axis.

Clearly, for C = 0, we obtain y = x2, a parabola with its vertex on the origin. The

curve y = x2 + 1 for C = 1 is obtained by shifting the parabola y = x2 one unit along

y-axis in positive direction. For C = – 1, y = x2 – 1 is obtained by shifting the parabola

y = x2 one unit along y-axis in the negative direction. Thus, for each positive value of C,

each parabola of the family has its vertex on the positive side of the y-axis and for

negative values of C, each has its vertex along the negative side of the y-axis.

### **CBSE Class 12 Commerce Mathematics Integrals:-****Comparison between differentiation and integration**

1. Both are operations on functions.

2. Both satisfy the property of linearity, i.e.,

(i) [ ] 1 1 ( ) 2 2 ( ) 1 1 ( ) 2 2 ( ) d k f x k f x k d f x k d f xdx dx dx+ = +

(ii) [ ] ∫ k1 f1 (x) + k2 f2 (x) dx = k1 ∫ f1 (x) dx + k2 ∫ f2 (x) dx

Here k1 and k2 are constants.

3. We have already seen that all functions are not differentiable. Similarly, all functions

are not integrable. We will learn more about nondifferentiable functions and

nonintegrable functions in higher classes.

4. The derivative of a function, when it exists, is a unique function. The integral of

a function is not so. However, they are unique upto an additive constant, i.e., any

two integrals of a function differ by a constant.

5. When a polynomial function P is differentiated, the result is a polynomial whose

degree is 1 less than the degree of P. When a polynomial function P is integrated,

the result is a polynomial whose degree is 1 more than that of P.

6. We can speak of the derivative at a point. We never speak of the integral at a

point, we speak of the integral of a function over an interval on which the integral

is defined as will be seen in Section 7.7.

7. The derivative of a function has a geometrical meaning, namely, the slope of the

tangent to the corresponding curve at a point. Similarly, the indefinite integral of

a function represents geometrically, a family of curves placed parallel to each

other having parallel tangents at the points of intersection of the curves of the

family with the lines orthogonal (perpendicular) to the axis representing the variable

of integration.

8. The derivative is used for finding some physical quantities like the velocity of a

moving particle, when the distance traversed at any time t is known. Similarly,

the integral is used in calculating the distance traversed when the velocity at time

t is known.

### **CBSE Class 12 Commerce Mathematics Integrals:-****Methods of Integration**

In previous section, we discussed integrals of those functions which were readily

obtainable from derivatives of some functions. It was based on inspection, i.e., on the

search of a function F whose derivative is f which led us to the integral of f. However,

this method, which depends on inspection, is not very suitable for many functions.

Hence, we need to develop additional techniques or methods for finding the integrals

by reducing them into standard forms. Prominent among them are methods based on:

1. Integration by Substitution

2. Integration using Partial Fractions

3. Integration by Parts

### **CBSE Class 12 Commerce Mathematics Integrals:-****Integration by substitution**

The given integral ∫ f (x) dx can be transformed into another form by changing

the independent variable x to t by substituting x = g (t).

### **CBSE Class 12 Commerce Mathematics Integrals:-****Integration by Partial Fractions**

Recall that a rational function is defined as the ratio of two polynomials in the form P( ) Q( ) x x , where P (x) and Q(x) are polynomials in x and Q(x) ≠ 0. If the degree of P(x) is less than the degree of Q(x), then the rational function is called proper, otherwise, it is called improper. The improper rational functions can be reduced to the proper rational

**“The integral of the product of two functions = (first function) × (integral**

**of the second function) – Integral of [(differential coefficient of the first function)**

**× (integral of the second function)]”**

**CBSE Class 12 Commerce Mathematics Integrals:-First fundamental theorem of integral calculus**

**Theorem 1** Let f be a continuous function on the closed interval [a, b] and let A (x) be

the area function. Then A′(x) = f (x), for all x ∈ [a, b].

### **CBSE Class 12 Commerce Mathematics Integrals:-****Second fundamental theorem of integral calculus**

We state below an important theorem which enables us to evaluate definite integrals

by making use of anti derivative.

**Theorem 2** Let f be continuous function defined on the closed interval [a, b] and F be

an anti derivative of f. Then ∫ ( ) b a f x dx = [F( )] = ba x F (b) – F(a).

### CBSE Class 12 Commerce Mathematics Integrals

**Recommended Articles**

CBSE Class 12 Commerce Mathematics Unit I Relations and Functions |

CBSE Class 12 Commerce Mathematics Matrices |

### CBSE Class 12 Commerce Mathematics Integrals

*CAKART provides India’s top class XI commerce faculty video classes – online Classes – at very cost effective rates. Get class XI commerce Video classes from CAKART.in to do a great preparation for your exam.*

*Watch class XI commerce Economics sample video lectures *

*Watch class XI commerce Accounting Sample video lecture Visit cakart.in*

*Watch class XI commerce Mathematics Sample video lecture Visit cakart.in*

*For any questions chat with us by clicking on the chat button below or give a missed call at 9980100288*