Integration - Calculus Notes | Study Engineering Mathematics - GATE

GATE: Integration - Calculus Notes | Study Engineering Mathematics - GATE

The document Integration - Calculus Notes | Study Engineering Mathematics - GATE is a part of the GATE Course Engineering Mathematics.
All you need of GATE at this link: GATE

Introduction

Let us consider Integration - Calculus Notes | Study Engineering Mathematics - GATE 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
Integration - Calculus Notes | Study Engineering Mathematics - GATE
Integration - Calculus Notes | Study Engineering Mathematics - GATE

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 =Integration - Calculus Notes | Study Engineering Mathematics - GATE
  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, Integration - Calculus Notes | Study Engineering Mathematics - GATE
Solution:
Derivative of √x is = 1 / 2√x
We use the substitution √x  = t
So, that Integration - Calculus Notes | Study Engineering Mathematics - GATE= dt ⇒ dx = 2t dt
Thus, Integration - Calculus Notes | Study Engineering Mathematics - GATE
= 2∫tan4 t sec2 t dt
Again we make another substitution tan t = u
sec2 t dt = dx
= 2∫ x4 dx = 2u5/ 5 + c
= Integration - Calculus Notes | Study Engineering Mathematics - GATE
[since u = tan t]
= Integration - Calculus Notes | Study Engineering Mathematics - GATE

Note:

Integration - Calculus Notes | Study Engineering Mathematics - GATE

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
Integration - Calculus Notes | Study Engineering Mathematics - GATE
= Integration - Calculus Notes | Study Engineering Mathematics - GATE
= Integration - Calculus Notes | Study Engineering Mathematics - GATE

Integrals of Some Particular Functions

  1.  Integration - Calculus Notes | Study Engineering Mathematics - GATE
  2.  Integration - Calculus Notes | Study Engineering Mathematics - GATE
  3. Integration - Calculus Notes | Study Engineering Mathematics - GATE
  4. Integration - Calculus Notes | Study Engineering Mathematics - GATE
  5.  Integration - Calculus Notes | Study Engineering Mathematics - GATE

To find the integral of type Integration - Calculus Notes | Study Engineering Mathematics - GATE, where p,q,a,b,c are constants we need to find real numbers A, B such that
Integration - Calculus Notes | Study Engineering Mathematics - GATE
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 Integration - Calculus Notes | Study Engineering Mathematics - GATE
Solution:  
Integration - Calculus Notes | Study Engineering Mathematics - GATE
Integration - Calculus Notes | Study Engineering Mathematics - GATE
=Integration - Calculus Notes | Study Engineering Mathematics - GATE

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) Integration - Calculus Notes | Study Engineering Mathematics - GATE
= x sin x – ∫sin x dx
= x sin x + cos x + c

Integrals of Some Particular Functions

  1. Integration - Calculus Notes | Study Engineering Mathematics - GATE
  2. Integration - Calculus Notes | Study Engineering Mathematics - GATE
  3. Integration - Calculus Notes | Study Engineering Mathematics - GATE
  4. Integration - Calculus Notes | Study Engineering Mathematics - GATE
  5. Integration - Calculus Notes | Study Engineering Mathematics - GATE
  6. Integration - Calculus Notes | Study Engineering Mathematics - GATE
  7. Integration - Calculus Notes | Study Engineering Mathematics - GATE
  8. Integration - Calculus Notes | Study Engineering Mathematics - GATE
  9. Integration - Calculus Notes | Study Engineering Mathematics - GATE
  10. Integration - Calculus Notes | Study Engineering Mathematics - GATE
  11. Integration - Calculus Notes | Study Engineering Mathematics - GATE

Definite Integrals

A real valued function f(x) is continuous on [a, b] thenIntegration - Calculus Notes | Study Engineering Mathematics - GATE is called definite integral. Geometrically it gives area of finite regions.  

Properties of Definite Integrals

  1. Integration - Calculus Notes | Study Engineering Mathematics - GATE
  2. Integration - Calculus Notes | Study Engineering Mathematics - GATE
  3. Integration - Calculus Notes | Study Engineering Mathematics - GATE
  4. Integration - Calculus Notes | Study Engineering Mathematics - GATE
  5. Integration - Calculus Notes | Study Engineering Mathematics - GATE
  6. Integration - Calculus Notes | Study Engineering Mathematics - GATE
  7. Integration - Calculus Notes | Study Engineering Mathematics - GATE
    = Integration - Calculus Notes | Study Engineering Mathematics - GATE where k = π/2 ; if m and n are even
    = 1 ; otherwise

Example 5:
Find the value of Integration - Calculus Notes | Study Engineering Mathematics - GATE
Solution:
Let, I = Integration - Calculus Notes | Study Engineering Mathematics - GATE
Also, I = Integration - Calculus Notes | Study Engineering Mathematics - GATE= Integration - Calculus Notes | Study Engineering Mathematics - GATE= Integration - Calculus Notes | Study Engineering Mathematics - GATE
Integration - Calculus Notes | Study Engineering Mathematics - GATE

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: Integration - Calculus Notes | Study Engineering Mathematics - GATE

Properties

  1. Γ(1) = 1 
  2. Γ(2) = 1     
  3. Γ(3) = 2
  4. Γ(n+1) = nΓ(n) = n!
  5. Integration - Calculus Notes | Study Engineering Mathematics - GATE
  6. Integration - Calculus Notes | Study Engineering Mathematics - GATE
  7. Integration - Calculus Notes | Study Engineering Mathematics - GATE

B. β – Function (Beta Function)
It is defined as β(m, n) = Integration - Calculus Notes | Study Engineering Mathematics - GATE

Properties

  1. β(m, n) = β(n, m) 
  2. β(m, n) =Integration - Calculus Notes | Study Engineering Mathematics - GATE
  3. β(m, n) =Integration - Calculus Notes | Study Engineering Mathematics - GATE
  4. Integration - Calculus Notes | Study Engineering Mathematics - GATE

Example 6:
Find the value of Integration - Calculus Notes | Study Engineering Mathematics - GATE
Solution:
Integration - Calculus Notes | Study Engineering Mathematics - GATE
2m - 1 = 1 / 2
⇒ m = 3 / 4
⇒ 2n - 1 = -1 / 2
⇒ n = 1 / 4
Integration - Calculus Notes | Study Engineering Mathematics - GATE

The document Integration - Calculus Notes | Study Engineering Mathematics - GATE is a part of the GATE Course Engineering Mathematics.
All you need of GATE at this link: GATE

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