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CBSE Class 12 Commerce Mathematics Integrals

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 Integrals

Differential 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

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