Integration - Calculus | Mathematics for NDA PDF Download

Introduction

Let us consider  We observe that the cos x is the derivative function of sin x or we can say that sin x is an anti-derivative (or an integral) of cos x.

Some Important Integrals

Some Properties of Indefinite Integrals

1. ∫ [f(x) + g(x)] dx = ∫f(x)dx + ∫g(x)dx
2. ∫kf(x)dx = k∫f(x)dx where k is constant
3. ∫f'(ax + b)dx =
4. ∫sinh x dx = cosh x + c
5. ∫cosh x dx = sinh x + c
6. ∫tanh x dx = log |cosh x| + c
7. ∫coth x dx = log |sinh x| + c
8. ∫sech x dx = tan-1 |sinh x | + c
9. ∫cosec2h x dx = –coth x + c
10. ∫cosech x  coth x = –cosech x + c
11. ∫sech x  tanh x dx = –sech x + c

Methods of Integration 

• Integration by substitution
• Integration using Trigonometric Identities
• Integration using partial fractions
• Integration by parts

1. 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)   Consider, I = ∫f(x)dx
Put, x = g(t) so that dx / dt = g'(t)
dx = g'(t)dt  
I = ∫f(x)dx = ∫f(g(t)) g'(t)dt

Example 1:
Integrate the following function,
Solution:
Derivative of √x is = 1 / 2√x
We use the substitution √x  = t
So, that = dt ⇒ dx = 2t dt
Thus,
= 2∫tan⁴ t sec² t dt
Again we make another substitution tan t = u
sec2 t dt = dx
= 2∫ x4 dx = 2u⁵/ 5 + c
=
[since u = tan t]
=

Note:

2. Integration using Trigonometric Identities
The following example can be solved by using trigonometric Identities  

Example 2:
∫cos2x dx
Solution:
We know,  cos 2x = 2 cos² x – 1

=
=

Integrals of Some Particular Functions

To find the integral of type , where p,q,a,b,c are constants we need to find real numbers A, B such that

To determine A and B we equate from both sides the coefficients of x and the constant terms. 

3. Integration by Partial Fractions
The following example can be solved by using partial fraction method  

Example 3:
Find
Solution:  


=

4. Integration by Parts
This can be done by using the given formula
f(x)g(x)dx = f(x)∫g(x)dx – ∫[∫g(x)dx] f'(x)dx
In integration by parts method proper choice of first and second function:
The first function is the word ILATE
I → Inverse Trigonometric
L → Logarithmic
A → Algebraic
T → Trigonometric
E → Exponential

Example 4:
Find the value of  ∫x cos x dx
Solution:
Put f(x) = x (first function) and g(x) = cos x (second function)
= x sin x – ∫sin x dx
= x sin x + cos x + c

Integrals of Some Particular Functions

Definite Integrals

A real valued function f(x) is continuous on [a, b] then is called definite integral. Geometrically it gives area of finite regions.  

Properties of Definite Integrals

1. 
2. 
3. 
4. 
5. 
6. 
7. 
   =  where k = π/2 ; if m and n are even
   = 1 ; otherwise

Example 5:
Find the value of
Solution:
Let, I =
Also, I = = =

Solving Indefinite Integrals Using Functions 
There are two functions which can be used to solve improper indefinite integrals. They are given as

A. Gamma Function
Gamma Function is defined by:

Properties

1. Γ(1) = 1 
2. Γ(2) = 1     
3. Γ(3) = 2
4. Γ(n+1) = nΓ(n) = n!
5. 
6. 
7. 

B. β – Function (Beta Function)
It is defined as β(m, n) =

Properties

1. β(m, n) = β(n, m) 
2. β(m, n) =
3. β(m, n) =
4. 

Example 6:
Find the value of
Solution:

2m - 1 = 1 / 2
⇒ m = 3 / 4
⇒ 2n - 1 = -1 / 2
⇒ n = 1 / 4