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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} )

2215_equation.JPG

                        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) = 304_equation.JPG, 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.

 

 1078_greatest integer function.JPG

  1. 33.                   158_integer.JPG is equal to

                        (A) 0                                                               (B) 2

                        (C) e                                                               (D) none of these 

Solution:       I = 158_integer.JPG

                         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 3x/3[x] dx is equal to (where [.] denotes greatest integer function) :

                      (A) 20                                                   (B) 40 / In3

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

2397_integer.JPG

                      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

1493_integer.JPG

                       2 ln √2  = ln 2.

                      Hence (A) is the correct answer.

 698_tan x, cot x.JPG

 

 

37.                 The value of 1370_integer.JPG  is equal to :

                      (A)  π/2                                                 (B) 2π

                      (C) π                                                     (D) π/p

Solution:     I = 2145_integer.JPG

                      Hence (B) is the correct answer.

38.                 The value of 481_integer.JPG is equal to :

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

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

Solution:     I = 481_integer.JPG  = |x–e–x|10 (1 - e-1) - (0 - 1) = 2 - e-1

                      Hence (A) is the correct answer.                                       

39.                 1868_integer.JPG has the value is :

                      (A) 0                                                      (B) 1/2           

                      (C) 1                                                     (D) 1/4

1125_integer.JPG

 Hence (A) is the correct answer.

40.                 1316_integer.JPG is :

                      (A) 0                                                      (B) 1

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

586_integer.JPG

                      Hence (D) is the correct answer. 

41.                 The value of 1076_integer.JPG depends on :

                      (A) p                                                      (B) q

                      (C) r                                                      (D) p and q

Solution:     I = 507_integer.JPG

                      = q 35_integer.JPG (Since sin3x and sin5 x are odd functions)

            Hence (B) is the correct answer.

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