Tiêu đề 21.INTEGRALS.pdf ✅

21. INTEGRALS -MCQS TYPE (1.) If the integral 5tan aln sin 2cos , tan 2 x dx x x x k x  = + − + − then a is equal to [AIEEE-2012] (a.) -2 (b.) 1 (c.) 2 (d.) -1 (2.) If  = f x dx x ( )  ( ) , then ( ) 5 3  x f x dx is equal to [JEE (Main)-2013] (a.) ( ) ( ) 1 3 3 2 3 3   x x x x dx C   −  +   (b.) ( ) ( ) 1 3 3 3 3 3 3 x x x x dx C   −  + (c.) ( ) ( ) 1 3 3 2 3 3 x x x x dx C   −  + (d.) ( ) ( ) 1 3 3 3 3 3   x x x x dx C   −  +   (3.) Let the population of rabbits surviving at a time t be governed by the differential equation ( ) ( ) 1 200 2 dp t p t dt = − . If p (0 100 ) = , then p t( ) equals [JEE (Main)-2014] (a.) 4/2 600 500 − e (b.) 12 400 300e − − (c.) /2 400 300 t − e (d.) 82 300 200e − − (4.) The integral ( ) 3/4 2 4 1 dx x x  + equals [JEE (Main)-2015] (a.) 1 4 4 4 x 1 c x   +   +   (b.) ( ) 1 4 4 x c + + 1 (c.) ( ) 1 4 4 − + + x c 1 (d.) 1 4 4 4 x 1 c x   + − +     (5.) The integral ( ) 12 9 3 5 3 2 5 1 x x x x +  + + is equal to [JEE (Main)-2016] (a.) ( ) 10 2 5 3 2 1 x C x x + + + (b.) ( ) 5 2 5 3 2 1 x C x x + + + (c.) ( ) 10 2 5 3 2 1 x C x x − + + + (d.) ( ) 5 2 5 3 1 x C x x − + + + where C is an arbitrary constant.
(6.) Let tan ,( 1) n n I xdx n =   . If 5 5 4 6 I I a x bx C + = + + tan , where C is a constant of integration, then the ordered pair (a b, ) is equal to [JEE (Main)-2017] (a.) 1 ,0 5       (b.) 1 , 1 5     −   (c.) 1 ,0 5     −   (d.) 1 ,1 5     −   (7.) The integral ( ) 2 2 2 5 3 2 3 2 5 sin cos sin cos sin sin cos cos x x dx x x x x x x  + + + is equal to [JEE (Main)-2018] (a.) ( ) 3 1 3 1 tan C x + + (b.) ( ) 3 1 3 1 tan C x − + + (c.) 3 1 1 cot C x + + (d.) 3 1 1 cot C x − + + (where C is a constant of integration) (8.) For 2 x n n N  +   1, (the set of natural numbers), the integral ( ) ( ) ( ) ( ) 2 2 2 2 2sin 1 sin2 1 2sin 1 sin2 1 x x x dx x x − − −  − + − is equal to (where c is a constant of integration) [JEE (Main)-2019] (a.) ( ) 1 2 log sec 1 2  − + x c (b.) 2 1 1 2 log sec 2 2 x c   −  +     (c.) 2 1 log cos 2 x c    −   +   (d.) ( ) 1 2 2 log sec 1 2 x c  − + (9.) If ( ) ( ) ( ) 8 6 2 2 7 5 7 , 0 1 2 x x f x dx x x x + =   + + , and f (0 0 ) = , then the value of f (1) is [JEE (Main)-2019] (a.) 1 2 (b.) 1 4 (c.) 1 2 − (d.) 1 4 − (10.) Let n  2 be a nutural number and 0 / 2     .
Then ( ) 1 1 sin sin cos sin n n d       + −  is equal to (where C is a constant of integration) [JEE (Main)-2019] (a.) 1 2 1 1 1 1 sin n n n n C n  + +     − + −   (b.) 1 2 1 1 1 1 sin n n n n C n  + −     − + +   (c.) 1 2 1 1 1 1 sin n n n n C n  + −     − + −   (d.) 1 2 1 1 1 1 sin n n n n C n  + −     + + −   (11.) If ( ) 3 3 5 4 4 1 48 x x x e dx e f x C − −  = + , where C is a constant of integration, then f x( ) is equal to [JEE (Main)-2019] (a.) 3 4 1 x + (b.) 3 − − 4 1 x (c.) 3 − + 2 1 x (d.) 3 − − 2 1 x (12.) If ( )( ) 2 2 4 1 1 m x dx A x x C x −  = − + , for a suitable chosen integer m and a function A x( ) , where C is a constant of integration, then ( ) ( ) m A x equals [JEE (Main)- 2019] (a.) 3 1 3x − (b.) 6 1 27x (c.) 4 1 9x (d.) 2 1 27x − (13.) If ( ) 1 2 1 2 1 x dx f x x C x +  = − + − , where C is a constant of integration, then f x( ) is equal to [JEE (Main)-2019] (a.) ( ) 1 1 3 x + (b.) ( ) 1 4 3 x + (c.) ( ) 2 4 3 x − (d.) ( ) 2 2 3 x + (14.) The integral  cos log ( e x dx ) is equal to (where C is a constant of integration) [JEE (Main)-2019] (a.) x x x C   cos log sin log ( 0 ) − + ( a )   (b.) cos log sin log ( ) ( ) 2 e e x   x x C + +   (c.) sin log cos log ( ) ( ) 2 e e x   x x C − +   (d.) x x x C   cos log sin log ( 0 ) + + ( e )   (15.) The integral ( ) 13 11 4 4 2 3 2 2 3 1 x x dx x x +  + + is equal to (where C is a constant of integration) [JEE (Main)-2019] (a.) ( ) 4 3 4 2 6 2 3 1 x C x x + + + (b.) ( ) 12 3 4 2 2 3 1 x C x x + + +
(c.) ( ) 12 3 4 2 6 2 3 1 x C x x + + + (d.) ( ) 4 3 4 2 2 3 1 x C x x + + + (16.) 5 sin 2 sin 2 x dx x  is equal to (where c is a constant of integration) [JEE (Main)-2019] (a.) 2 sin 2sin2 x x x c + + + (b.) x x x c + + + 2sin 2sin2 (c.) x x x c + + + 2sin sin2 (d.) 2 sin sin2 x x x c + + + (17.) If ( ) ( )( ) 1 6 3 3/3 3 6 1 1 dx xf x x C x x  = + + + where C is a constant of integration, then the function f x( ) is equal to: [JEE (Main)-2019] (a.) 3 1 2x − (b.) 2 3 x (c.) 3 1 6x − (d.) 2 1 2x − (18.) The integral 2/3 4/3 sec cosec x xdx is equal to (Here C is a constant of integration) [JEE (Main)-2019] (a.) 1/3 3cot x C − − + (b.) 1/3 3tan x C − − + (c.) 3 4/3 tan 4 x C − − + (d.) 1/3 3tan x C − + (19.) If ( ( ) )) sec 2 sec tan sec tan sec x  + + e x xf x x x x dx ( ) secx = + e f x C , then a possible choice of f x( ) is [JEE (Main)-2019] (a.) 1 sec tan 2 x x − − (b.) 1 sec tan 2 x x + + (c.) 1 sec tan 2 x x x + − (d.) 1 sec tan 2 x x t x + + (20.) If ( ) 2 2 2 10 dx x x  − + ( ) 1 2 1 tan 3 2 10 x f x A C x x −     − = + +       − +   where C is a constant of integration, then [JEE (Main)-2019] (a.) 1 81 A = and f x x ( ) = − 3 1 ( ) (b.) 1 54 A = and f x x ( ) = − 3 1 ( ) (c.) 1 27 A = and f x x ( ) = − 9 1 ( ) (d.) 1 54 A = and ( ) 2 f x x = − 9( 1)