Pascal's Triangle & Binomial Theorem

Definite Integral

MD. MOSFIQUR RAHMAN

Senior Lecturer in Mathematics Dept. of GED,FSIT

Integration

Definition: The process of finding anti-derivative is called integration

The expression is called the indefinite integral

∫ ^{� ( � ) �� = � ( � ) + �}

Type:

Integration

Indefinite Integration Definite Integration

Definite Integral

Definite Integral: If is a continuous function defined in the interval then the definite integral with respect to is defined as,

∫

�

�

� ( � ) �� = [ ^{� ( � )} ]

_�^�

^{= � ( � ) − � ( �)} ∫

�

�

� ( � ) ��

Integrand

(function we want to integrate)

Integral Symbol

Integral with respect to Upper limit of

Integration

Lower limit of Integration

This formula is known as Newton-Leibnitz formula

Where and are called lower and upper limits of integration, respectively

Note:

• The indefinite integral is a function of , whereas definite integral is a number.

• Given we can find given we cannot find .

Geometrical Interpretation

Integration can be used to find areas, volumes, central points and many useful things.

But it is often used to find the area under the graph of a function like this:

∫

�

�

� ( � ) ��

Definite Integral (from a to b)

�

�

A ^{� ( �)}

� �

∫ ^{� ( � ) ��}

Indefinite Integral (no specific values)

A ^{� ( �)}

�

�

Concept of Definite Integral

∫

1 2

2 � ��=? Example:

Step 1: we need to find the indefinite integral

Using the rules of integration we find that

Step 3: calculate that at 1 and 2:

At : At :

Step 4: Subtract:

(value at upper limit – value at lower limit)

( ₂

²

_{+ �} ) ₋ ( ₁

²

_{+ �} )

2

²

+ � − 1

²

− �

4 − 1 + � − � =3

And “C” gets cancelled out… so with Definite Integrals we can ignore C.

2

Step 2: use limit for definite integral:

∫

1 2

2 � ��=

[

^�2+�

]

₁²

Properties of Definite Integral

• The definite integral of 1 is equal to the length of interval of the integral

∫

�

�

1 �� =� − �

• A constant factor can be moved across the integral sign

∫

�

�

� . � (� ) �� = � . ∫

�

�

� ( � ) ��

• Definite integral is independent of variable of integration

∫

�

�

� ( � ) �� = ∫

�

�

� ( � ) � �

• If the upper limit and the lower limit of a definite integral are the same, then the integral is zero

�

Example : ∫

−1 3

1 �� =3 − (− 1 )=4

Example : ∫

0 1

2 �

²

�� =2 ∫

0 1

�

²

��

Example : ∫

1 2

�

²

�� = ∫

1 2

�

²

� �

Example : ∫

1

�

²

�� 0

Properties of Definite Integral

• Reversing the limit of integration change the sign of definite integral

∫

�

�

� ( � ) �� =− ∫

�

�

� ( � ) ��

• If c is any point in the interval then the definite integral of over is equal to the sum of integrals over and

∫

�

�

� ( � ) �� = ∫

�

�

� ( � ) � � + ∫

�

�

� ( � ) � �

∫

0

�

� (�)��=∫

0

�

� (�− �) ��

∫

�

� (�)��=

{

^{� 0 ,�h�� � ( �)�� ���}

Example : ∫

1 2

�

²

�� =− ∫

2 1

�

²

��

Example : ∫

1 3

�

²

�� = ∫

1 2

�

²

�� + ∫

2 3

�

²

��

Problem 1:

 

Given that,

∫

−3 1

(

6 �²+5 �−2

)

_��

¿ [ ^{6 � 3}

³

^{+5 � 2}

²

^{− 2 �} ]

−3 1

¿ ( ^{2 × 1}

³

^{+ 5 1 2}

²

^{− 2 × 1} ) ⁻ ( ^{2 × ( − 3 )}

³

^{+ 5 ( − 2 3 )}

²

^{− 2 × ( − 3 )} )

¿

(

^{2+ 52 − 2}

)

⁻

(

^{−54 + 4 52 +6}

)

^{=52 − 5 12}

¿ 5

2 − 5 1

2 =− 46

1 2

Problem 2:

 

Given that,

∫

0 log 2

�

^�

1 + �

^�

��

¿

[

^log

⁽

^1+��

⁾ ]

0 log 2

¿ log ( _{1+ �}

^log2

) _{− log} ( _{1 + �}

⁰

)

¿ log ( 1 + 2 ) − log ( 1 + 1 )

¿ log 3 − log 2

¿log 3 2

∴

log 2

�

^�

�� 3