TRB BT Assistant - Evaluation of Definite Integrals Online Test (MCQ Quiz)

1. 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

2. 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

3. 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)

4. 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

5. 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)

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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|

12. 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

13. 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

14. 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

15. 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

16. 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)

17. 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

18. What is the value of ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> 1 / (x² + 1) dx?

A) 0 B) π C) 2π D) 1/2

19. Evaluate ∫₀²<0xC2><0x80><0x93> dθ / (2 + cos θ).

A) π / √3 B) 2π / √3 C) π D) 2π

20. 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

21. 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

22. 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)

23. 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

24. 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

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. What is the value of ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> x³ / (x⁴ + 1) dx?

A) 0 B) π C) 2π D) 1

27. 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)

28. 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

29. 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

30. What is the value of ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> dx / (x⁴ + 1)?

A) π / √2 B) 2π / √2 C) π D) 2π

31. 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

32. 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

33. 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

34. 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

35. 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

36. 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

37. 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

38. 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

39. 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

40. 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

41. 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

42. 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

43. 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

44. 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| → ∞

45. 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

46. 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

47. 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

48. 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

49. 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

Evaluation of Definite Integrals - Online Test

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