Tiêu đề 21. Definite Inegrals Hard .pdf ✅

1.)The value of the integral  − 2[x] 0 (x [x])dx is - (a.) [x] (b.) 2 1 [x] (c.) 3[x] (d.) 2[x] 2.)For x  R and a continuous function ƒ, let I1 =  1+cos t sin t 2 2 x ƒ {(2 – x)} dx and I2 =  1+cos t sin t 2 2 ƒ{x(2 – x)} dx. Then I1/I2 is - (a.) 0 (b.) 1 (c.) 2 (d.) 3 3.)Evaluate :  b a x | x | dx, a < b - (a.) |a| – |b| (b.) |b| + |a| (c.) |b| – |a| (d.) None of these 4.)Given  + 1 0 1 t sin t dt = , find the value of   −  + − 4 4 2 4 2 t sin(t / 2) in terms of  - (a.)  (b.) –  (c.) 2 (d.) None of these 5.)Evaluate :   3 16 0 sin x dx (a.) 4 21 (b.) 2 21 (c.) 2 11 (d.) 4 11 6.) Show that  + 1 0 (1 x) ln x dx = –  + 1 0 x ln (1 x) dx = – 12 2  (a.) 12 2 
(b.) 6 2  (c.) – 6 2  (d.) – 12 2  7.)Evaluate :  − 1 0 1 cot (1 – x + x2 ) dx - (a.) 2  – ln 2 (b.) 2  – ln 4 (c.) 4  – ln 2 (d.) None 8.)Evaluate :  − +  + 1 1 [x[1 sin x] 1] dx, where [.] is the greatest integer function - (a.) 2 (b.) 1 (c.) 3 (d.) None of these 9.)The value of x 0 lim → x cost dt x 0 2  is – (a.) 1 (b.) 0 (c.) – 1 (d.) 2 10.)F(x) =  + x 0 (3sin u 4cosu) du on the interval (5/4, 4/3] is - (a.) 3/2 – 3 /2 (b.) 2 5 − 4 3 (c.) 2 7 − 4 3 (d.) 2 9 − 4 3 11.)For any t  R and ƒ a continuous function, let I1 =  + − 1 cos t sin t 2 2 xƒ(x(2 x))dx and I2 =  + − 1 cos t sin t 2 2 ƒ(x(2 x))dx then I1/I2 is equal to- (a.) 2
(b.) 1 (c.) 4 (d.) None of these 12.)Evaluate :  − 102 0 1 [tan x] dx, where [·] denotes the greatest integer function less than or equal to x - (a.) 102 – tan 2 (b.) 102 – tan 1 (c.) 101 – tan 2 (d.) None of these 13.)If ƒ : R→R, g : R→R are continuous functions then the value of the integral   − + − − / 2 / 2 [(ƒ(x) ƒ( x))(g(x) – g( x))]dx is - (a.) 1 (b.) 0 (c.) –1 (d.)  14.)Given f an odd function periodic with period 2 continuous  x and g(x) =  x 0 ƒ(t)dt , then - (a.) g(x) is odd function (b.) g(2n) = 1 (c.) g(2n) = 0 (d.) None of these 15.) Let f(x) be positive, continuous and differentiable on the interval (a, b) and → + x a lim f(x) = 1, – x b lim → f(x) = 31/4 . If f (x)  f 3 (x) + f(x) 1 then the greatest value of b – a is - (a.) 1 (b.) 3 1/4 (c.) (31/4 – 1) (d.) 4  16.)Let G(x, t) =    −  −  t(x 1),when t x and tis continuousfunction of x in [0,1] x(t 1),when x t If g(x) =  1 0 f(t) G(x, t)dt, then which is incorrect (a.) g(0) + g(1) = 0 (b.) g(0) = 0 (c.) g(1) = 1 (d.) g  (x) = f(x)
17.)I2 =  + − 1 cos t sin t 2 2 f{x(2 x)}dx.Then 2 1 I I is - (a.) 0 (b.) 1 (c.) 2 (d.) 3 18.)   − → 4 2 x 0 x 0 1 2 x 0 sin t dt (tan t) dt lim is equal to - (a.) 1 (b.) –1 (c.) –1/2 (d.) 1/2 19.)If ƒ(x) =  + 3 x 1 4 1 t dt , then ƒ(x) is equal to - (a.) 12 2 12 (1 x ) 6x(1 5x ) + − (b.) 12 2 12 (1 x ) 6x(1 5x ) + + (c.) – 12 2 12 (1 x ) 6x(1– 5x ) + (d.) None of these 20.)Let dx d F(x) =         x e sin x , x > 0 If  4 1 x 3 3 sin x e dx = F(k) – F(1), then one of the possible values of k is - (a.) 15 (b.) 16 (c.) 63 (d.) 64 21.) n→ lim         + + + + + + 8n 1 ... (n 2) n (n 1) n n 1 3 2 3 2 is equal to – (a.) 8 3 (b.) 4 1 (c.) 8 1 (d.) None of these