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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} )

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.                    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 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

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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