Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation

CA Foundation: Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation

The document Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation is a part of the CA Foundation Course Business Mathematics and Logical Reasoning & Statistics.
All you need of CA Foundation at this link: CA Foundation


INTEGRAL CALCULUS
INTEGRATION

Integration is the reverse process of differentiation.
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
We know
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integration is the inverse operation of differentiation and is denoted by the symbol  .

Hence, from equation (1), it follows that
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
i.e. Integral of xn with respect to variable x is equal to Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Thus if we differentiate Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation  we can get back xn.
Again if we differentiate Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation and c being a constant, we get back the same xn
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Hence Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation 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 Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
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 Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Examples : 
Find
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Solution:
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation where c is arbitrary constant
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Examples: Evaluate the following integral:
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation

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 Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
We put (2x + 3) = t ⇒ so 2 dx = dt or dx = dt / 2
Therefore
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
This method is known as Method of Substitution
Example:
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
We put (x2 +1) = t
so 2x dx = dt or x dx = dt / 2
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation

IMPORTANT STANDARD FORMULAE
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Examples:
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation where z=ex dz = ex dx
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation

INTEGRATION BY PARTS
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
where u and v are two different functions of x
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation

METHOD OF PARTIAL FRACTION
Type I:
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
[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 Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation

Type II:
Example
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Solution:
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation

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 Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
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 Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation

Example:
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Solution:
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation  we put x3 = z, 3x2 dx = dz
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Example: Find the equation of the curve where slope at (x, y) is 9x and which passes through the origin.
Solution:
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
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 Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation

‘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 Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Solution: 
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Note: In definite integration the constant (c) should not be added
Example:
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Solution: 
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Now,
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation

IMPORTANT PROPERTIES
Important Properties of definite Integral
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Example:
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Solution: 
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Example: Evaluate Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Solution:
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
let x5 = t so that 5x4 dx = dt
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation (by standard formula b)
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation
Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation

The document Integral Calculus Notes | Study Business Mathematics and Logical Reasoning & Statistics - CA Foundation is a part of the CA Foundation Course Business Mathematics and Logical Reasoning & Statistics.
All you need of CA Foundation at this link: CA Foundation

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