Tiêu đề 4-integral-calculus-2.pdf ✅

4. Integral Calculus - 2 1. If 2∫0 1 tan−1 xdx = ∫0 1 cot−1 (1 − x + x 2 )dx, then ∫0 1 tan−1 (1 − x + x 2 )dx is equal to : (A) log 4 (B) π 2 + log 2 (C) log 2 (D) π 2 − log 4 2. The area (in sq. units) of the region described by A = {(x, y) ∣ y ≥ x 2 − 5x + 4, x + y ≥ 1, y ≤ 0} is : (A) 7 2 (B) 19 6 (C) 13 6 (D) 17 6 3. The value of the integral ∫4 10 [x 2 ]dx [x 2−28x+196]+[x 2] , where [x] denotes the greatest integer less than or equal to x, is (A) 6 (B) 3 (C) 7 (D) 1/3 4. For x ∈ R, x ≠ 0, if y(x) is a differential function such that x∫1 x y(t)dt = (x + 1)∫1 x ty(t)dt, then y(x) equals : (where C is a constant.) (A) C x e − 1 x (B) C x 2 e − 1 x (C) C x 3 e − 1 x (D) Cx 3e 1 x 5. For x > 0, let f(x) = ∫1 x log t 1+t dt. Then f(x) + f ( 1 x ) is equal to : (A) 1 4 (log x) 2 (B) 1 2 (log x) 2 (C) log x (D) 1 4 log x 2 6. The area (in square units) of the region bounded by the curves y + 2x 2 = 0 and y + 3x 2 = 1, is equal to : (A) 3/5 (B) 3/4 (C) 1/3 (D) 4/3 7. Let f: R → R be a function such that f(2 − x) = f(2 + x) and f(4 − x) = f(4 + x), for all x ∈ R and ∫0 2 f(x)dx = 5. Then the value of ∫10 50f(x)dx is : (A) 80 (B) 100 (C) 125 (D) 200 8. Let f: (−1,1) → R be a continuous function. If ∫0 sin x f(t)dt = √3 2 x, then f ( √3 2 ) is equal to : (A) √3 2 (B) √3 (C) √ 3 2 (D) 1 2 9. The integral ∫0 1 2 ln (1+2x) 1+4x 2 dx, equals : (A) π 4 ln 2 (B) π 8 ln 2 (C) π 16 ln 2 (D) π 32 ln 2 10. If for n ≥ 1,Pn = ∫1 e (log x) ndx, then P10 − 90P8 is equal to : (A) -9 (B) 10e (C) −9e (D) 10 11. If [ ] denotes the greatest integer function, then the integral ∫0 π [cos x]dx is equal to : (A) − π 2 (B) 0
(C) π 2 (D) -1 12. If for a continuous function f(x), ∫−π t (f(x) + x)dx = π 2 − t 2 , for all t ≥ −π, then f (− π 3 ) is equal to : (A) π (B) π/3 (C) π/2 (D) π/6 13. Let function F be defined as F(x) = ∫1 x e t t dt, x > 0 then the value of the integral ∫1 x e t t+a dt, where a > 0, is : (A) e a [F(x) − F(1 + a)] (B) e −a [F(x + a) − F(a)] (C) e a [F(x + a) − F(1 + a)] (D) e −a [F(x + a) − F(1 + a)] 14. If x = ∫0 y dt √1+t 2 , then d 2y dx 2 is equal to : (A) y (B) √1 + y 2 (C) y/√1 + y 2 (D) y 2 15. The area bounded by the curve y = ln (x) and the lines y = 0, y = ln (3) and x = 0 is equal to : (A) 3 (B) 3In (3) − 2 (C) 3In (3) + 2 (D) 2 16. The area of the origin region (in sq. units), in the quadrant, bounded by the parabola y = 9x 2 and the lines x = 0, y = 1 and y = 4, is : (A) 7/9 (B) 14/3 (C) 7/3 (D) 14/9 17. The integral ∫7π/4 7π/3 √tan2 xdx is equal to : (A) log 2√2 (B) log 2 (C) 2log 2 (D) log √2 18. The value of ∫−π/2 π/2 sin2 x 1+2 x dx is : (A) π (B) π/2 (C) 4π (D) π/4 19. The area under the curve y = |cos x − sin x|,0 ≤ x ≤ π 2 , and above x-axis is : (A) 2√2 (B) 2√2 − 2 (C) 2√2 + 2 (D) 0 20. The value of ∫−1 2 {2x}dx (where function {.} denotes fractional part function) is : (A) 1/2 (B) 1 (C) 3/2 (D) 0 21. The value of ∫0 10π (|sin x| + |cos x|)dx is : (A) 20 (B) 40 (C) 30 (D) None of these 22. If a ≤ ∫1 3 √(3 + x 3) ≤ b then value of a and b are : (A) a = 3, b = √30 (B) a = 3, b = 2√30 (C) a = 4, b = 2√30 (D) None of these 23. The value of limn→∞ 3 n [1 + √ n n+3 + √ n n+6 + √ n n+9 + ⋯ + √ n n+3(n−1) ] is : (A) π (B) − π 2 (C) π 2 (D) None of these 24. Let f(x) = Maximum {x 2 , (1 − x) 2 , 2x(1 − x)} where 0 ≤ x ≤ 1. The area of the region bounded by the curves y = f(x), x-axis, x = 0 and x = 1 is : (A) 15/27 square units (B) 17/27 square units (C) 15/14 square units (D) None of these 25. The area bounded by the y-axis and the curve x = e y sin πy, y = 0, and y = 1 is : (A) (e+1)π 1+π2 (B) (e+1) 1+π2
(C) (e+1)π 1+π (D) None of these 26. If f(x) = min{x + 1, √1 − x}, then the value of ∫−1 1 12 7 f(x)dx. (A) 2 (B) -2 (C) 1 (D) None of these 27. The area of the region bounded by y = {x} and 2x − 1 = 0, y = 0, ({} stands for fraction part of x ) (A) 1/2 (B) 1/4 (C) 1/8 (D) None of these 28. If f(x) = { x x < 1 x − 1 x ≥ 1 , then ∫0 2 x 2f(x)dx is equal to : (A) 1 (B) 4/3 (C) 5/3 (D) 5/2 29. If f(0) = 1, f(2) = 3, f ′ (2) = 5 and f ′ (0) is finite, then ∫0 1 x ⋅ f ′′(2x)dx is equal to : (A) Zero (B) 1 (C) 2 (D) None of these 30. The value of ∫−1 3 (|x − 2| + [x])dx is ([x] stands for greatest integer less than or equal to x ) (A) 7 (B) 5 (C) 4 (D) 3 31. The value of ∫lnπ−ln 2 ln π e x 1−cos ( 2 3 e x) dx is equal to : (A) √3 (B) −√3 (C) 1 √3 (D) − 1 √3 32. If I1 = ∫e e 2 dx lnx and I2 = ∫1 2 e x x dx, then : (A) I1 = I2 (B) 2I1 = I2 (C) I1 = 2I2 (D) None of these 33. Let I1 = ∫0 3π f(cos2 x)dx,I2 = ∫0 2π f(cos2 x)dx and I3 = ∫0 π f(cos2 x)dx, then : (A) I1 + 2I3 + 3I2 = 0 (B) I1 = 2I2 + I3 (C) I2 + I3 = I1 (D) I1 = 2I3 34. The value of ∫0 ∞ x (1+x)(1+x 2) dx equal to : (A) π/4 (B) π/2 (C) is same as ∫0 ∞ dx (1+x)(1+x 2) (D) Cannot be evaluated 35. The value of integral ∫a b |x| x dx, a < b is : (A) b − a if 0 < a < b (B) a − b if a < b < 0 (C) b + a if a < 0 < b (D) |b| − |a| 36. Let f: R → R, g: R → R be continuous functions. Then the value of integral ∫ln λ ln 1/λ f( x 2 4 )[f(x)−f(−x)] g( x2 4 )[g(x)+g(−x)] dx is : (A) Depends on λ (B) A non-zero constant (C) Zero (D) None of these 37. ∫2−ln3 3+ln ln (4+x) ln (4+x)+ln (9−x) dx is equal to : (A) Cannot be evaluate (B) is equal to 5/2 (C) is equal to 1 + 2ln 3 (D) is equal to 1/2 + ln 3 38. The value of integral ∫0 π xf(sin x)dx is : (A) π 2 ∫0 π f(sin x)dx (B) π∫0 π/2 f(sin x)dx (C) 0 (D) None of these 39. If f(x) = ∫0 x (cos4 t + sin4 t)dt, then f(x + π) is equal to : (A) f(x) + f(π) (B) f(x) + 2f(π) (C) f(x) + f ( π 2 ) (D) f(x) + 2f ( π 2 )
40. The value of limh→0 ∫a x+h ln 2 tdt−∫a x ln 2 tdt h equals to : (A) 0 (B) ln 2x (C) 2ln x x (D) Does not exist 41. The value of x, f(x) = 1 + x + ∫1 x (ln 2 t + 2lnt)dt, where f ′ (x) vanishes is : (A) e −1 (B) 0 (C) 2e −1 (D) 1 + 2e −1 42. If ∫0 y cos t 2dt = ∫0 x 2 sin t t dt, then the value of dy dx is : (A) 2sin2 x xcos2 y (B) 2sin x 2 xcos y2 (C) 2sin x 2 x(1+2sin y2 2 ) (D) None of these 43. The value of limn→∞ π n [sin π n + sin 2π n + ⋯ . . +sin (n−1)π n ] is : (A) 0 (B) π (C) 2 (D) None of these 44. limn→∞ [(1 + 1 n2 ) (1 + 2 2 n2 ) ... (1 + n 2 n2 )] 1/n is equal to : (A) e π/2 2e 2 (B) 2e 2e π/2 (C) 2 e 2 e π/2 (D) None of these 45. The value of limn→∞∑r=1 n ( r 3 r 4+n4 ) is : (A) log 2 (B) 1 2 log 2 (C) 1 3 log 2 (D) 1 4 log 2 46. The area bounded by curve y 3 − 9y + x = 0 and y-axis is : (A) 9 2 (B) 9 (C) 81 2 (D) 81 47. The area bounded by y = 2 − |2 − x| and y = 3 |x| is : (A) 4+3ln 3 2 (B) 4−3ln 3 2 (C) 3 2 + ln3 (D) 1 2 + ln3 48. For which of the following values of m, is the area of the region bounded by the curve y = x − x 2 and the line y = mx equals 9 2 ? (A) -4 (B) -2 (C) 2 (D) 4 49. The area of the region bounded by x = 0, y = 0, x = 2, y = 2, y ≤ e x and y ≥ lnx, is : (A) 6 − 4ln2 (B) 4ln 2 − 2 (C) 2ln 2 − 4 (D) 6 − 2ln2 50. The area between two arms of the curve |y| = x 3 from x = 0 to x = 2 is : (A) 2 (B) 4 (C) 8 (D) 16 51. Statement 1 : If {.} represents fractional part function, then∫0 5.5 {x}dx = 21 8 . Statement 2 : If [.] and {.} represents greatest integer and fractional part functions respectively, then ∫0 t {x}dx = [t] 2 + {t} 2 2 (A) Statement-1 is True, Statement-2 is True and Statement-2 is a correct explanation for Statement-1 (B) Statement-1 is True, Statement-2 is True and Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True 52. ∫0 2nπ (|sin x| − [| sin x 2 |]) dx (where [ ] denotes the greatest integer function and n ∈ I ) is equal to : (A) 0