Solved Examples of Definite Integral

Solved Examples

30.The value of  ∫¹⁰⁰₀(√x)dx ( where {x} is the fractional part of x) is

(A) 50                                                         

(B) 1

(C) 100

(D) none of these

Solution:       Given integral = ∫¹⁰⁰₀ (√x–[√x])dx               ( by the def. of {x} )

                        Hence (D) is the correct answer.                          

31.                   The value of  ∫¹₀   (|sin 2 p x| dx is equal to

                        (A) 0                                                          (B)  2/π

                        (C) 1/π                                                      (D) 2

Solution:       Since |sin 2 π x | is periodic with period 1/2,

                        I =  ∫¹₀  |sin 2 π x| dx= 2 ∫¹₀  sin 2 π x dx

                        = 2 [–cos2πx/2π]^1/2₀ = 2/π 

                        Hence (B) is the correct answer.

32.                   Let f : R —> R, f(x) = , where [.] denotes greatest integer function, then ∫⁴_–2 f(z)dx is equal to

                        (A) 5/2                                                            (B) 3/2

                        (C) 5                                                               (D) 3

Solution:       x – [x] = {x}                         x – [x +1] ={x} – 1                         ∫4–2 f(x)dx = 6.1/2 (1.1) = 3                         Hence (D) is the correct answer.

33. 33.                    is equal to

                        (A) 0                                                               (B) 2

                        (C) e                                                               (D) none of these 

Solution:       I = 

                         property ∫^a_–a f(x)dx = 0 (f (–x) = –f (x), odd function)

                        Hence I = 0

                        Hence (A) is the correct answer.

34.                 The value of ∫¹⁰_–10 3^x/3^[x] dx is equal to (where [.] denotes greatest integer function) :

                      (A) 20                                                   (B) 40 / In3

                      (C) 20 / In 3                                          (D) none of these

                      Hence (B) is the correct answer. 

35.                 Values of ∫^+1/2_–1/2 cos x log 1+x/1–x dx is :

                      (A) 1/2                                                  (B) – 1/2

                      (C) 0                                                     (D) none of these 

Solution:     I = ∫^+1/2_–1/2 cos x log 1+x/1–x dx

                      f (x) = cos x ln 1+x/1–x

                      f (- x) = cox (- x) ln 1+x/1–x

                      = - cos (x) ln (1+x/1–x) = – f (x)

                      f (x) is an odd function 

                      hence I = 0

                      Hence (C) is the correct answer. 

36.                 f (x) = min (tan x, cot x), 0 < x < , then ∫^π/2₀ f(x)dx is equal to :

                      (A) ln2                                                  (B) ln √2

                      (C) 2 ln √2                                           (D) none of these

Solution:     f (x) = min (tan x, cot x),                               ∈ [0, π/2]                       f (x) = tan x,    0 < x <  π/4                              = cot x,    π/4  < x < π/4                       Hence                        2 ln √2  = ln 2.                       Hence (A) is the correct answer.

37.                 The value of   is equal to :

                      (A)  π/2                                                 (B) 2π

                      (C) π                                                     (D) π/p

Solution:     I = 

                      Hence (B) is the correct answer.

38.                 The value of  is equal to :

                      (A) 2 – 1/e                                  (B) 2 + 1/e

                      (C) e+1/e                                   (D) none of these

Solution:     I =   = |x–e^–x|¹₀ (1 - e^-1) - (0 - 1) = 2 - e^-1

                      Hence (A) is the correct answer.                                       

39.                  has the value is :

                      (A) 0                                                      (B) 1/2           

                      (C) 1                                                     (D) 1/4

 Hence (A) is the correct answer.

40.                  is :

                      (A) 0                                                      (B) 1

                      (C) π/2                                                  (D) π/4

                      Hence (D) is the correct answer. 

41.                 The value of  depends on :

                      (A) p                                                      (B) q

                      (C) r                                                      (D) p and q

Solution:     I = 

                      = q  (Since sin³x and sin⁵ x are odd functions)

            Hence (B) is the correct answer.

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