Evaluation of Definite Integrals - Question Bank

1. The evaluation of ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> x^m / (x^n + 1) dx (where n > m+1) typically involves:
A) A unit circle contour
B) A semi-circular contour
C) A sector contour
D) A rectangular contour
2. Consider the integral ∫₀<0xE2><0x88><0x9E> x^(p-1) dx / (1 + x²) for 0 < p < 2. This integral can be evaluated using a contour that includes:
A) A circular arc
B) A semi-circular arc
C) A keyhole contour or a sector contour
D) A rectangular contour
3. The value of the integral ∫₀²<0xC2><0x80><0x93> dθ / (5 - 4 cos θ) is:
A) 2π / 3
B) π / 3
C) 2π
D) π
4. When evaluating ∫₀²<0xC2><0x80><0x93> dθ / (5 - 4 cos θ), the corresponding complex integral is taken over:
A) A large circle |z| = R
B) The unit circle |z| = 1
C) A semi-circle in the upper half-plane
D) A rectangular contour
5. For the integral ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> f(x)e^(iax) dx with a > 0, Jordan's Lemma states that the integral over the upper semi-circular arc vanishes if:
A) f(z) → 0 uniformly as |z| → ∞
B) f(z) = O(1/|z|) as |z| → ∞
C) f(z) = O(1) as |z| → ∞
D) f(z) is analytic in the upper half-plane
6. The condition for the integral along the arc of a semi-circle to vanish as R → ∞, for ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> f(x) dx, is typically:
A) f(z) → 0 uniformly as |z| → ∞
B) f(z) = O(1/|z|²) as |z| → ∞
C) f(z) = O(1/|z|) as |z| → ∞
D) f(z) is bounded as |z| → ∞
7. Which type of integral is often evaluated by transforming it into a contour integral using z = e^(iθ)?
A) Improper integrals over the real line
B) Integrals of rational functions of trigonometric functions
C) Integrals involving exponential functions
D) Integrals of logarithms
8. What is the value of ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> x² / (x⁴ + 4) dx?
A) π / √2
B) π
C) 2π / √2
D) 2π
9. The technique of 'deformation of contours' is used when:
A) The function is analytic everywhere
B) The integration path needs to be simplified
C) Poles lie on the contour of integration
D) The integral is over an infinite domain
10. If a function f(z) is analytic everywhere except for a finite number of poles, the integral around any simple closed contour C is given by:
A) 2πi times the sum of residues inside C
B) 0 if there are no poles inside C
C) The sum of the function values inside C
D) The derivative of the function at the poles inside C
11. Consider the integral ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> e^(iax) / (x - x₀) dx, where x₀ is real and a > 0. The choice of contour depends on:
A) The value of x₀
B) The sign of 'a'
C) The value of the integral
D) The type of pole
12. What is the value of the principal value integral P.V. ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> 1/x dx?
A) 0
B) π
C) 2π
D) Undefined
13. For an integral of the form ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> f(x) dx, if f(z) has poles on the real axis, a common technique is to use:
A) A large circular contour
B) A semi-circular contour with indentation
C) A keyhole contour
D) Cauchy's Integral Formula
14. Which of the following integrals CANNOT be directly evaluated using the standard Residue Theorem with simple semi-circular or circular contours?
A) ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> dx / (x² + 1)
B) ∫₀²<0xC2><0x80><0x93> dθ / (2 + cos θ)
C) ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> x / (x³ - 1) dx
D) ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> sin(x) / x dx
15. The integral ∫₀<0xE2><0x88><0x9E> dx / (x⁶ + 1) can be evaluated using a semi-circular contour. The poles are roots of z⁶ + 1 = 0. The residues in the upper half-plane are needed.
A) Sum of residues in upper half-plane
B) Twice the sum of residues in upper half-plane
C) Half the sum of residues in upper half-plane
D) Sum of residues in lower half-plane
16. Evaluating ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> x² / (x⁴ + 1) dx involves finding poles of z⁴ + 1 = 0. These poles are:
A) ±1, ±i
B) ±(1+i)/√2, ±(1-i)/√2
C) ±i, ±(1+i)/√2
D) ±1, ±(1+i)/√2
17. What is the value of ∫₀²<0xC2><0x80><0x93> ln(sin θ) dθ?
A) 0
B) -π log 2
C) π log 2
D) -2π log 2
18. The integral ∫₀²<0xC2><0x80><0x93> ln(sin θ) dθ can be evaluated using complex analysis by considering:
A) A standard unit circle contour
B) A semi-circular contour
C) A contour involving the logarithm function
D) A specific substitution z = e^(iθ) and relating ln(sin θ) to complex logs
19. Consider the integral ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> x / (x³ - 1) dx. This integral requires indenting the contour because:
A) There are poles at infinity
B) There are poles on the real axis
C) The function is not rational
D) The degree condition is not met
20. What is the value of ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> dx / (x⁴ + 1)?
A) π / √2
B) 2π / √2
C) π
D) 2π
21. If the condition deg(Q) ≥ deg(P) + 1 holds for ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> P(x)/Q(x) dx, and Q(x) has no real roots, the integral is:
A) 0
B) 2πi * Sum of residues in upper half-plane
C) πi * Sum of residues in upper half-plane
D) Sum of residues in upper half-plane
22. For the integral ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> P(x)/Q(x) dx, where P and Q are polynomials with deg(Q) ≥ deg(P) + 2, the integral is related to:
A) Sum of residues of P(z)/Q(z) in the upper half-plane
B) Sum of residues of P(z)/Q(z) in the lower half-plane
C) Twice the sum of residues of P(z)/Q(z) in the upper half-plane
D) Half the sum of residues of P(z)/Q(z) in the upper half-plane
23. The integral ∫₀<0xE2><0x88><0x9E> x^(m-1) / (1 + x) dx can be evaluated using a keyhole contour. The value is:
A) π / sin(mπ)
B) 2π / sin(mπ)
C) π / cos(mπ)
D) 2π / cos(mπ)
24. What is the value of ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> x³ / (x⁴ + 1) dx?
A) 0
B) π
C) 2π
D) 1
25. When evaluating ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> e^(ix) / (x - i) dx, the pole is at z = i. Which contour is used?
A) Upper semi-circle (pole is outside)
B) Lower semi-circle (pole is inside)
C) Unit circle
D) Rectangular contour
26. Consider ∫₀²<0xC2><0x80><0x93> cos(2θ) / (1 + cos²θ) dθ. The appropriate contour and substitution lead to an integral whose poles are found by solving:
A) z² + 1 = 0
B) z² - 2z + 1 = 0
C) z² + 2z + 1 = 0
D) z² + 2z - 1 = 0
27. The integral ∫₀<0xE2><0x88><0x9E> dx / (x² + 1) is evaluated using a semi-circular contour. Its value is:
A) π
B) 2π
C) π/2
D) 1
28. What is the value of ∫₀²<0xC2><0x80><0x93> dθ / (a + b cos θ), where a > |b|?
A) 2π / √(a² - b²)
B) π / √(a² - b²)
C) 2π / (a² - b²)
D) π / (a² - b²)
29. For the integral ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> cos(x) / (x² + a²) dx (a > 0), the value is:
A) π / a
B) 2π / a
C) πe⁻ᵃ / a
D) 2πe⁻ᵃ / a
30. The integral ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> sin(x) / x dx is a classic example. Using complex analysis, its value is:
A) 0
B) π
C) 2π
D) π/2
31. Evaluate ∫₀²<0xC2><0x80><0x93> dθ / (2 + cos θ).
A) π / √3
B) 2π / √3
C) π
D) 2π
32. What is the value of ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> 1 / (x² + 1) dx?
A) 0
B) π
C) 2π
D) 1/2
33. Consider the integral ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> x / (x² + 1) dx. What is the value of this integral?
A) 0
B) π
C) 2π
D) 1
34. When indenting a contour around a simple pole z₀ on the real axis, the contribution of the indentation to the integral is:
A) 0
B) πi * Res(f, z₀)
C) 2πi * Res(f, z₀)
D) -πi * Res(f, z₀)
35. If a function f(z) has poles on the real axis, how is the integral ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> f(x) dx typically evaluated?
A) The integral diverges
B) The contour is indented around the poles
C) The poles on the real axis are ignored
D) The integral is evaluated using a unit circle
36. For the integral ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> f(x)e^(iax) dx with a < 0, which contour and lemma are typically used?
A) Upper semi-circle and Jordan's Lemma
B) Lower semi-circle and Jordan's Lemma
C) Unit circle and Cauchy's Theorem
D) Rectangular contour and Residue Theorem
37. If f(z) satisfies the conditions of Jordan's Lemma (upper half-plane, a > 0), and has a simple pole at z₀ in the upper half-plane, then:
A) ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> f(x)e^(iax) dx = 2πi * sum of residues in upper half-plane
B) ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> f(x)e^(iax) dx = 2πi * sum of residues in lower half-plane
C) ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> f(x)e^(iax) dx = πi * sum of residues in upper half-plane
D) ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> f(x)e^(iax) dx = 0
38. Jordan's Lemma is particularly useful for evaluating integrals involving:
A) Trigonometric functions
B) Exponential functions like e^(iaz) where a > 0
C) Rational functions
D) Polynomials
39. When using a semi-circular contour in the upper half-plane for ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> f(x) dx, the integral along the arc tends to zero as the radius R tends to infinity if:
A) |f(z)| ≤ M/|z|² for large |z|
B) |f(z)| ≤ M/|z| for large |z|
C) |f(z)| ≤ M for large |z|
D) |f(z)| tends to infinity for large |z|
40. For integrals of the form ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> f(x) dx, which contour is typically used?
A) A large circle centered at the origin
B) A semi-circle in the upper half-plane
C) A rectangular contour
D) A keyhole contour
41. Consider the integral ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> f(x) dx. If f(x) is an even function, it can be related to the integral over a semi-circular contour by:
A) ∫<0xE2><0x88><0x9E>₀ f(x) dx = (1/2) ∫<0xE1><0xB5><0x9C> f(z) dz
B) ∫<0xE2><0x88><0x9E>₀ f(x) dx = ∫<0xE1><0xB5><0x9C> f(z) dz
C) ∫<0xE2><0x88><0x9E>₀ f(x) dx = 2 ∫<0xE1><0xB5><0x9C> f(z) dz
D) ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> f(x) dx = ∫<0xE1><0xB5><0x9C> f(z) dz
42. The integral ∫₀²<0xC2><0x80><0x93> R(cos θ, sin θ) dθ is related to the contour integral ∫<0xE1><0xB5><0x9C> g(z) dz where C is the unit circle, by:
A) ∫<0xE1><0xB5><0x9C> g(z) dz = ∫₀²<0xC2><0x80><0x93> R(cos θ, sin θ) dθ
B) ∫<0xE1><0xB5><0x9C> g(z) dz = i ∫₀²<0xC2><0x80><0x93> R(cos θ, sin θ) dθ
C) ∫<0xE1><0xB5><0x9C> g(z) dz = (1/i) ∫₀²<0xC2><0x80><0x93> R(cos θ, sin θ) dθ
D) ∫<0xE1><0xB5><0x9C> g(z) dz = 2πi ∫₀²<0xC2><0x80><0x93> R(cos θ, sin θ) dθ
43. For the substitution z = e^(iθ) on the unit circle, cos θ and sin θ can be expressed in terms of z as:
A) cos θ = (z + z⁻¹)/2, sin θ = (z - z⁻¹)/(2i)
B) cos θ = (z - z⁻¹)/2, sin θ = (z + z⁻¹)/(2i)
C) cos θ = (z + z⁻¹)/(2i), sin θ = (z - z⁻¹)/2
D) cos θ = (z - z⁻¹)/(2i), sin θ = (z + z⁻¹)/2
44. When using the unit circle contour C: |z|=1 for real integrals, the substitution z = e^(iθ) implies dz = i e^(iθ) dθ, which means:
A) dθ = dz / (iz)
B) dθ = dz / z
C) dθ = iz dz
D) dθ = dz
45. Which contour is commonly used to evaluate integrals of the form ∫₀²<0xC2><0x80><0x93> f(cos θ, sin θ) dθ?
A) Rectangular contour
B) Semi-circular contour
C) Keyhole contour
D) Circular contour (unit circle)
46. If f(z) has a pole of order m at z₀, the residue can be calculated using:
A) lim(z→z₀) (z - z₀)ᵐ f(z)
B) (m-1)! / lim(z→z₀) (z - z₀)ᵐ f(z)
C) 1 / (m-1)! * lim(z→z₀) dᵐ⁻¹/dzᵐ⁻¹ [(z - z₀)ᵐ f(z)]
D) lim(z→z₀) f(z) / (z - z₀)ᵐ⁻¹
47. For a function f(z) with a simple pole at z₀, the residue is given by:
A) lim(z→z₀) f(z)
B) lim(z→z₀) z f(z)
C) lim(z→z₀) (z - z₀) f(z)
D) lim(z→z₀) f'(z) / (z - z₀)
48. The Residue Theorem relates the integral of a function around a closed curve to:
A) The function's values inside the curve
B) The sum of the residues of its poles inside the curve
C) The derivative of the function inside the curve
D) The number of zeros of the function inside the curve
49. Which theorem is fundamental for evaluating definite integrals using complex analysis?
A) Cauchy's Integral Theorem
B) Residue Theorem
C) Maximum Modulus Principle
D) Argument Principle
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