Pascal's Triangle & Binomial Theorem

Integration (Part – 1)

Shirin Sultana Lecturer

Department of GED

Daffodil International University

2

Concept of Integration

Any operation in mathematics has an inverse operation such as Addition Subtraction

Multiplication Division (function) (inverse function)

(matrix) (inverse matrix)

 

The reverse process of differentiation is defined as anti-differentiation or simply integration

 

 

Integral

 

�

²

   

2 �

Derivative

Concept of Integration (cont.…)

∫ � ( � ) ��

 

Integrand

(function we want to integrate)

Integral Symbol Integral with respect to

 

∫ � ( � ) �� = � ( � ) + �

 

Why “+ C” ???

Constant of integration

Integral

2 �

 

 

 

�

²

   

∫ 2 ��� = � 2 + �

sometimes abosorbed into integrand can be written as

can be written as

 

4

Integration Formulas

Source: Calculus, 10th Edition. Author: Howard Anton, Irl C. Bivens, Stephen Davis, page. 324.

Rule 1:

 

∫ 4 ��� � �� = 4 ∫ ��� ���

 

∫

 

������ =sin �

¿ 4 sin � + �

 

Example: E

 

∫ ( � + � 2 ) �� = ∫ ��� + ∫ � 2 ��

 

∫ �

^�

�� = �

�+1

�+ 1 , � ≠ − 1

 

¿ �

¹⁺¹

1+ 1 + �

²⁺¹

2 + 1 + �

 

Example: E

 

Rule 2:

 

�

²

�

³

 

Rule 3:

 

6

∫ � ( � 2 + � − 1 ) �� = ∫ ( � 3 + � 2 − � ) ��

 

∫ �

^�

�� = �

�+1

� + 1 , � ≠ − 1

¿ ∫ � 3 �� + ∫ � 2 �� − ∫ ���

 

 

Example 1: E

 

¿ �

⁴

4 + �

³

3 − �

²

2 + �

 

Example 2: E

 

∫ ( � � − 5 � � + 2 ) �� = ∫ � � �� − ∫ 5 � � �� + ∫ 2 ��

 

∫ �� = �

¿ � � − 5 ∫ � � �� + 2 ∫ ��

 

 

¿ �^� −5 �^�

ln � +2 �+�

 

∫ �

^�

�� = �

�

ln �

 

∫ � � �� = � �

 

Example 3: E

 

∫ � + 2 √ � + 7

√ � �� = ∫ ( √ � � + 2 √ �

√ � +

7

√ � ) �� = ∫ ( �

¹²

+ 2 + 7 �

⁻¹²

) ��

 

¿ ∫ �

1

2

�� + ∫ 2 �� + ∫ 7 �

⁻

1

2

��

 

¿ �

1 2+1

1

2 +1

+2 � +7 �⁻

1 2+1

− 1

2 + 1

+�

 

¿ �

3 2

3 +2 � +7 �

1 2

1 + �

 

¿ 2

3 ( √ � )

³

+ 2 � + 14 √ � + �

 

∫ �

^�

�� = �

�+1

� + 1

 

Prove that

 

8

¿ ln | � | + �

 

∫

 

1 � �� = ln | � |

Proof

Suppose  

� ( � ) = �

�

�� [ � ( � ) ] = �� � ( � )

 

�

^′

( � ) = � �

��

 

∴ �

^′

( � ) �� = ��

 

∫ �

′

( � )

� ( � ) �� = ∫ 1 � � �

 

So,

¿ ln | � ( � ) | + �

 

∴ ∫ �

′

( � )

� ( � ) �� =�� | � ( � ) | +�

 

Prove that

 

¿

 

− ∫ − cos sin � � ��

Proof

∫

 

tan ��� = ∫ cos sin � � � �

Now,

∴ ∫ tan ��� =�� | sec � | + �

 

¿ − �� | cos � | + �

 

¿ −��

|

^sec1 �

|

^+�

 

¿ − ( �� | 1 | − �� | sec � |) + �

 

¿ − ( 0 − �� | sec � |) + �

 

Exercise: Prove that

 

∫ �

′

( � )

� ( � ) �� =�� | � ( � ) | + �

 

Prove that

 

10

¿ ∫ cosec

2

� −cosec � cot �

( cosec � − cot � ) ��

 

Proof

∫ cosec � �� = ∫ cosec � ( cosec � − cot � )

( cosec � − cot � ) � �

 

Now,

∴

 

∫ ����� ��� =�� | ����� � − ��� � | + �

¿ ∫ −cosec � cot � + cosec

2

�

( cosec � − cot � ) ��

 

Exercise: Prove that

 

 

∫ �

′

( � )

� ( � ) �� =�� | � ( � ) | + �

¿ �� | cosec � − cot � | + �