This document discusses definite integration from a course on advanced mathematics. It defines definite integration, lists elementary properties of definite integrals including basic formulas and properties involving multiple integrals. It also covers standard properties related to inequalities and the mean value theorem. Examples are provided to demonstrate evaluating definite integrals of basic functions between limits using properties and formulas. The document concludes with exercises asking to evaluate definite integrals of various functions.

1. Definite Integration
Course- Diploma
Semester-II
Subject- Advanced Mathematics
Unit- V
RAI UNIVERSITY, AHMEDABAD

2. Unit-V Definite Integration
RAI UNIVERSITY, AHMEDABAD
5. Definite integration
5.1 Definition— If ( ) is continuous in the interval [ , ] and ( ) is an antiderivative of
( ), then
5.2 Elementary properties of definite integral—
1. ∫ ( ) = − ∫ ( )
2. ∫ ( ) = ∫ ( ) = ∫ ( )
3. ∫ ( ) =
4. ∫ ( ) = ∫ ( ) + ∫ ( ) , < <
5. ∫ ( + − ) = ∫ ( )
6. ∫ ( − ) = ∫ ( )
7. ∫ ( ) =
∫ ( )
8. ∫ ( ) =
∫ ( ) ( − ) = ( )
( − ) = − ( )
5.3 Standard properties of definite integral—
1. If ( ) ≥ is defined and continuous on[ , ], then ∫ ( ) ≥
2. If ( ) ≤ is defined and continuous on[ , ], then ∫ ( ) ≤
3. If ( ) ≤ ( ) is defined and continuous on[ , ], then ∫ ( ) ≤ ∫ ( )
( ) = ( ) − ( )

3. Unit-V Definite Integration
RAI UNIVERSITY, AHMEDABAD
4. If ( ) and ( ) be two functions defined and continuous on [a, b] then—
( ) ( ) ≤ ( ) ( )
5. The mean value theorem—
Statement— If ( ) is continuous on [ , ], then at least one value of = in ( , )
such that—
( ) = ∫ ( )
Where ( ) is called as average or mean value of ( ) on [ , ].
Example— Integrate ∫ .
Solution—∫
= = − =
Example— Integrate ∫
Solution—∫
= |− cos |
= − cos — cos 0
= 0 − (−1) = 1
Example— Integrate ∫
Solution— ∫
= ∫ − ∫
( )
(∫ )
= − ∫
= [ − ∫ ]
= [ − ]
= ( − ) − ( 1 − 1)
= ( − ) + 1 = 1
Example— Evaluate ∫
Solution—∫ 3 2
= ∫ 2 3 2
= ∫ ( 5 + )

4. Unit-V Definite Integration
RAI UNIVERSITY, AHMEDABAD
= − 5 −
= − 5 − − − 0 − 0
= (0 − 0) − − − 1
=
EXERCISE
Question— Evaluate ∫ sin 4 cos 3 .
Question— Integrate ∫ (2 + 1) .
Question— Integrate ∫ cos .
Question— Integrate ∫ ( ) .

5. Unit-V Definite Integration
RAI UNIVERSITY, AHMEDABAD
Reference—
1. mathworld.wolfram.com
2. www.mathsisfun.com
3. tutorial.math.lamar.edu
4. en.wikipedia.org/wiki/Integral
5. https://www.khanacademy.org
6. www.sosmath.com