Integral Calculus CA CPT Notes | EduRev

Business Mathematics and Logical Reasoning & Statistics

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CA CPT : Integral Calculus CA CPT Notes | EduRev

The document Integral Calculus CA CPT Notes | EduRev is a part of the CA CPT Course Business Mathematics and Logical Reasoning & Statistics.
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INTEGRAL CALCULUS
INTEGRATION

Integration is the reverse process of differentiation.
Integral Calculus CA CPT Notes | EduRev
We know
Integral Calculus CA CPT Notes | EduRev
Integration is the inverse operation of differentiation and is denoted by the symbol  .

Hence, from equation (1), it follows that
Integral Calculus CA CPT Notes | EduRev
i.e. Integral of xn with respect to variable x is equal to Integral Calculus CA CPT Notes | EduRev
Thus if we differentiate Integral Calculus CA CPT Notes | EduRev  we can get back xn.
Again if we differentiate Integral Calculus CA CPT Notes | EduRev and c being a constant, we get back the same xn
Integral Calculus CA CPT Notes | EduRev
Hence Integral Calculus CA CPT Notes | EduRev and this c is called the constant of integration

Integral calculus was primarily invented to determine the area bounded by the curves dividing the entire area into infinite number of infinitesimal small areas and taking the sum of all these small areas.

BASIC FORMULAS
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev
Note: In the answer for all integral sums we add +c (constant of integration) since the differentiation of constant is always zero.

Elementary Rules:
Integral Calculus CA CPT Notes | EduRev
Examples : 
Find
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev
Solution:
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev where c is arbitrary constant
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev
Examples: Evaluate the following integral:
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev

METHOD OF SUBSTITUTION (CHANGE OF VARIABLE)
It is sometime possible by a change of independent variable to transform a function into another which can be readily integrated.
We can show the following rules.
To put z = f (x) and also adjust dz = f'(x) dx

Example: ∫F{ h(x )} h'(x ) dx, take ez = h(x) and to adjust dz = h'(x) dx

then integrate F(z) d using normal rule.
Example: Integral Calculus CA CPT Notes | EduRev
We put (2x + 3) = t ⇒ so 2 dx = dt or dx = dt / 2
Therefore
Integral Calculus CA CPT Notes | EduRev
This method is known as Method of Substitution
Example:
Integral Calculus CA CPT Notes | EduRev
We put (x2 +1) = t
so 2x dx = dt or x dx = dt / 2
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev

IMPORTANT STANDARD FORMULAE
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev
Examples:
Integral Calculus CA CPT Notes | EduRev where z=ex dz = ex dx
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev

INTEGRATION BY PARTS
Integral Calculus CA CPT Notes | EduRev
where u and v are two different functions of x
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev

METHOD OF PARTIAL FRACTION
Type I:
Integral Calculus CA CPT Notes | EduRev
[Here degree of the numerator must be lower than that of the denominator; the denominator contains non–repeated linear factor]
so 3x + 2 = A (x – 3) + B (x – 2)
We put x = 2 and get
3.2 + 2 = A (2–3) + B (2–2) => A = –8
we put x = 3 and get
3.3 +2 = A (3–3) + B (3–2) => B= 11
Integral Calculus CA CPT Notes | EduRev

Type II:
Example
Integral Calculus CA CPT Notes | EduRev
Solution:
Integral Calculus CA CPT Notes | EduRev

or 3x + 2 = A (x – 2) (x – 3) + B (x – 3) +C (x – 2)2
Comparing coefficients of x2, x and the constant terms of both sides, we find
A+C = 0 …………(i)
–5A + B – 4C = 3 ……(ii)
6A – 3B + 4C = 2 …….(iii)
By (ii) + (iii) A – 2B = 5 ..…….(iv)
(i) – (iv) 2B + C = –5 …….(v)
From (iv) A = 5 + 2B
From (v) C = –5 – 2B
From (ii) –5 ( 5 + 2B) + B – 4 (– 5 – 2B) = 3
or – 25 – 10B + B + 20 + 8B = 3
or – B – 5 = 3
or B = – 8, A = 5 – 16 = – 11, from (iv) C = – A = 11
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev
Type III:

so 3x2 –2x +5 = A (x2 + 5 ) + (Bx +C) (x–1)

Equating the coefficients of x2, x and the constant terms from both sides we get
A + B = 3 …………(i)
C – B = –2 …………(ii)
5A – C = 5 ………….(iii)
by (i) + (ii) A + C = 1 ……… (iv)
by (iii) + (iv) 6A = 6 ……… (v)
or A = 1
therefore B = 3 – 1 = 2 and C = 0
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev

Example:
Integral Calculus CA CPT Notes | EduRev
Solution:
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev  we put x3 = z, 3x2 dx = dz
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev
Example: Find the equation of the curve where slope at (x, y) is 9x and which passes through the origin.
Solution:
Integral Calculus CA CPT Notes | EduRev
Since it passes through the origin, c = 0; thus required eqn . is 9x2 = 2y.

DEFINITE INTEGRATION
Suppose F(x) dx = f (x)
As x changes from a to b the value of the integral changes from f (a) to f (b). This is as
Integral Calculus CA CPT Notes | EduRev

‘b’ is called the upper limit and ‘a’ the lower limit of integration. We shall first deal with

indefinite integral and then take up definite integral.
Example:
Integral Calculus CA CPT Notes | EduRev
Solution: 
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev
Note: In definite integration the constant (c) should not be added
Example:
Integral Calculus CA CPT Notes | EduRev
Solution: 
Integral Calculus CA CPT Notes | EduRev
Now,
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev

IMPORTANT PROPERTIES
Important Properties of definite Integral
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev
Example:
Integral Calculus CA CPT Notes | EduRev
Solution: 
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev
Example: Evaluate Integral Calculus CA CPT Notes | EduRev
Solution:
Integral Calculus CA CPT Notes | EduRev
let x5 = t so that 5x4 dx = dt
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev (by standard formula b)
Integral Calculus CA CPT Notes | EduRev
Integral Calculus CA CPT Notes | EduRev

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