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Integral Calculus Lecture-1 Integration Dated:-27.07.2020 PPT-23 UG (B.Sc., Part-2) Dr. Md. Ataur Rahman Guest Faculty Department of Mathematics M.L. Arya, College, Kasba PURNEA UNIVERSITY, PURNIA
Contents 1. Definition of integration 2. Indefinite integral 3. Rules of integration 4. Methods of integration 5. Integration of some special functions 6. Definite integral 7. Improper integral 8. Integration as the limit of a sum 9. Reduction formulae 10. Application of integration
Integration (Anti-differentiation) Definition (1):-Integration is the inverse process of differentiation. i.e. ( ) ( ), F x f x then dx Integrand or derivative or differential coefficient d = = ( ) f x dx ( ) F x Integral or Primitive or anti- derivative of f(x) (2) The process of finding the integral (anti- derivative) whose derivative is given is called integration.
Indefinite integral d dx We know that ( ) = , Sinx Cosx then d dx ( ) = + = , , Cosxdx Sinx Also Sinx C Cosx = + then Cosxdx Sinx C i.e. Integrals of cosx are Sinx and Sinx +C Here Sinx is called particular integral and Sinx+C is called general integral (called indefinite integral) Note:- Indefinite integral contains an arbitrary constant C, is called constant of integration
Rules of Indefinite integral Rule I:- If u is a function of x, then where k is a constant. Examples (1) 9 9 x dx x dx = ( ) u v w dx + + + k udx 2 = , kudx 3 x = = + 2 3 9 3 x C 3 = = + (2) 5 5 5 Cosxdx Cosxdx Sinx C Rule II:- Where u, v, w, are functions of x. Examples (1) 4 x + = + + + udx vdx wdx 4 3 2 x x x ( ) + + = + + + x C + = + + + x C + 3 2 4 3 2 3 2 5 4 3 2 5 5 x x dx x x x 4 3 2 1 x 1 x 1 x + + = + + = + + 2 2 (2) sec cos sec sin tan Cosx x dx x x x x C 2 2
Rules of Indefinite integral ( ) ( ) ( ) dx d f x d dx Rule III:- where k is a constant. Example ( dx = = + ( ) f x dx ( ) f x and ( ) f x dx C = + (1) Cosxdx Sinx C ) d d dx ( ) = + = + = cos 0 cos Cosxdx Sinx C x x Again, d dx ( ) = cos sin x x d dx ( ) ( ) = = x c + cos sin cos x dx x dx
Methods of integration 1. Method of transformation 2. Method of substitution 3. Integration by parts 4. Integration by partial fraction
Method of transformation Type I Algebraic functions:- Express the given function (expression) as the sum or difference following forms by using the formulae of Algebra ( ) 2 (1). (2). 1 1 1 1 (4). (5). or a x x a x x 1 1 + 1 1 n + n (3). x or ax b or or + 2 2 x a x 2 2 2 1 x a x 1 x 1 + (6). or 2 2 2 2 ax b 2 1 1 + 1 + a a b b a a b b Tricks:- = = 1. a b a b
Problems Integrate the following w. r. t. + + 2 1 + 4 3 3 x x x x + (1). (2). (3). + 2 + + + 3 4 3 x x 3 x 5 8 3 4 x x ( )( 2 ) 3 1 x + + + 2 5 4 3 1 1 5 6 9 x x x x x (4). (5). (6). 2 2 4 4 x x x Soln (1):- + + + + 1 1 3 3 5 5 3 3 4 4 x x x x = dx dx + + + + + + 3 5 3 4 3 5 3 4 x x x x + + 3 3 5 5 3 + 3 x 4 x x x 1 1 ( ) ( ) = = + + 3 5 3 4 dx x x dx 2 2 4 2 3 2 3 3 3 ( ) ( ) = + + + 3 5 3 4 x x C 2 2
Solution ( )( 2 )( ) ) ( + + + + 3 2 8 1 2 2 4 ( 1) x x + x x x x Solution (4):- = dx dx 2 2 4 2 4 x x x x 3 2 x x ( ) = + = + x C + 2 2 2 x x dx 3 2 + 5 4 3 5 6 9 6 x x x x Solution (5):- = + 4 5 9 dx x x dx 4 x 2 3 2 x x x = + + = + + + 3 5 6log 9 5 6log 3 x x C x x x C 2 3 2
