Tiêu đề 1. INDIFINITE INTEGRAL - 1-18.pmd.pdf ✅

NISHITH Multimedia India (Pvt.) Ltd., 1 JEE MAINS - CW - VOL - I JEE ADVANCED - VOL - VII INDEFINITE INTEGRATION NISHITH Multimedia India (Pvt.) Ltd., SYNOPSIS Integration of Irrational Algebraic functions 1. To evaluate 2 . ( )r dx x k ax bx c     we substitue, x k t   1/ . Thus substitution will reduce the given integral in to 1 2 r t dt At Bt c     2. To evaluate 2 2 ( ) ( ) fx ax bx c dx dx e gx h       we write, 2 1 1 1 ax bx c A dx e dx e C       ( )(2fx+g)+B ( ) Where 1 1 A B, and C1 are constants which can be obtained by comparing the coefficient of like terms on both sides and given integral will re- duce to the form 1 1 1 2 2 2 (2fx+g) fx fx ( ) fx dx dx A dx B C gx h gx h dx e gx h             3. To evaluate 2 2 2 2 2 2 ( ) ( ) , , ( ) ( ) ( ) ax bx c dx ax bx c dx ax bx c ex fx g dx ex fx g ex fx g                put 2 2 ax bx c A ex fx g B ex f c         ( ) (2 ) Find the values of A,B and C by comparing the coefficient of 2 x x, and constant term. 4. To evaluate 2 ( ) ( ) ax b dx cx e ex fx g      we put ( ) ( ) ax b A cx e B     . Find the values of A and B by comparing the coefficient of x and con- stant term. 5. To evaluate 2 2 ( ) ( ) ax bx c dx ex f gx hx i       we put 2 ax bx c A ex f gx h B ex f c         ( )(2 ) ( ) Find the values of A,B and C by comparing the coefficient of 2 x x, and constant term. 6. To evaluate 2 2 ( ) ( ) xdx ax b cx e    we put 2 2 cx e t   Integration of the Functions of the Type m n p x (a+bx ) : Case- I: If p N  (Natural number), we expand the integral with the help of binomial theorem and integrate Case - II : If p I  , write k x = p , where k is the L.C.M of denominators of m and n. Case - III : If m+1 n is an integer, we put n (a+bx ) k  r , where k is the denominator of the fraction p. Case - IV : If m+1 n  p is an integer, we put , n k n a bx r x   where k is the denominator of the fraction p. 7. Integrals of the form 2 R x ax bx c dx ( , )    are calculated with the aid of one of the three Euler susbstitutions. 1. 2 ax bx c t x a if a      0; 2. 2 ax bx c tx c if c      0; 3. 2 2 ax bx c x tif ax bx c a x x          ( ) ( )( )    i.e.,if  is a real root of 2 ax bx c    0 . LEVEL-V SINGLE ANSWER QUESTIONS 1. If x f t    cos sin , sin cos t f t t y f t t f t t              then 1 2 2 2 dx dy dt dt dt                      (A)     ' f t f t C   (B)     '' f t f t C   (C)     ''' ' f t f t C   (D)     '' ' f t f t C   INDEFINITE INTEGRATION