Evaluation of Definite Integrals - One Line Questions
1.
Evaluating ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> x² / (x⁴ + 1) dx involves finding poles of z⁴ + 1 = 0. These poles are: —
±(1+i)/√2, ±(1-i)/√2
2.
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: —
|f(z)| ≤ M/|z|² for large |z|
3.
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: —
∫<0xE1><0xB5><0x9C> g(z) dz = (1/i) ∫₀²<0xC2><0x80><0x93> R(cos θ, sin θ) dθ
4.
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: —
∫<0xE2><0x88><0x9E>₀ f(x) dx = (1/2) ∫<0xE1><0xB5><0x9C> f(z) dz
5.
Which of the following integrals CANNOT be directly evaluated using the standard Residue Theorem with simple semi-circular or circular contours? —
∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> x / (x³ - 1) dx
6.
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: —
∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> f(x)e^(iax) dx = 2πi * sum of residues in upper half-plane
7.
When indenting a contour around a simple pole z₀ on the real axis, the contribution of the indentation to the integral is: —
πi * Res(f, z₀)
8.
Consider the integral ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> x / (x² + 1) dx. What is the value of this integral? —
0
9.
What is the value of ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> 1 / (x² + 1) dx? —
π
10.
The integral ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> sin(x) / x dx is a classic example. Using complex analysis, its value is: —
π
11.
What is the value of ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> x³ / (x⁴ + 1) dx? —
0
12.
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: —
2πi * Sum of residues in upper half-plane
13.
What is the value of ∫₀²<0xC2><0x80><0x93> ln(sin θ) dθ? —
-π log 2
14.
What is the value of the principal value integral P.V. ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> 1/x dx? —
0
15.
What is the value of ∫₀²<0xC2><0x80><0x93> dθ / (a + b cos θ), where a > |b|? —
2π / √(a² - b²)
16.
The value of the integral ∫₀²<0xC2><0x80><0x93> dθ / (5 - 4 cos θ) is: —
2π / 3
17.
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: —
2πi times the sum of residues inside C
18.
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 keyhole contour or a sector contour
19.
When evaluating ∫₀²<0xC2><0x80><0x93> dθ / (5 - 4 cos θ), the corresponding complex integral is taken over: —
The unit circle |z| = 1
20.
For integrals of the form ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> f(x) dx, which contour is typically used? —
A semi-circle in the upper half-plane
21.
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 semi-circular contour with indentation
22.
The integral ∫₀²<0xC2><0x80><0x93> ln(sin θ) dθ can be evaluated using complex analysis by considering: —
A specific substitution z = e^(iθ) and relating ln(sin θ) to complex logs
23.
The evaluation of ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> x^m / (x^n + 1) dx (where n > m+1) typically involves: —
A sector contour
24.
Which theorem is fundamental for evaluating definite integrals using complex analysis? —
Residue Theorem
25.
For the substitution z = e^(iθ) on the unit circle, cos θ and sin θ can be expressed in terms of z as: —
cos θ = (z + z⁻¹)/2, sin θ = (z - z⁻¹)/(2i)
26.
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: —
dθ = dz / (iz)
27.
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: —
f(z) = O(1/|z|²) as |z| → ∞
28.
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: —
f(z) → 0 uniformly as |z| → ∞
29.
Which type of integral is often evaluated by transforming it into a contour integral using z = e^(iθ)? —
Integrals of rational functions of trigonometric functions
30.
If f(z) has a pole of order m at z₀, the residue can be calculated using: —
1 / (m-1)! * lim(z→z₀) dᵐ⁻¹/dzᵐ⁻¹ [(z - z₀)ᵐ f(z)]
31.
For a function f(z) with a simple pole at z₀, the residue is given by: —
lim(z→z₀) (z - z₀) f(z)
32.
Which contour is commonly used to evaluate integrals of the form ∫₀²<0xC2><0x80><0x93> f(cos θ, sin θ) dθ? —
Circular contour (unit circle)
33.
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. —
Half the sum of residues in upper half-plane
34.
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: —
Twice the sum of residues of P(z)/Q(z) in the upper half-plane
35.
The technique of 'deformation of contours' is used when: —
Poles lie on the contour of integration
36.
The Residue Theorem relates the integral of a function around a closed curve to: —
The sum of the residues of its poles inside the curve
37.
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? —
The contour is indented around the poles
38.
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: —
The sign of 'a'
39.
Consider the integral ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> x / (x³ - 1) dx. This integral requires indenting the contour because: —
There are poles on the real axis
40.
Jordan's Lemma is particularly useful for evaluating integrals involving: —
Exponential functions like e^(iaz) where a > 0
41.
When evaluating ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> e^(ix) / (x - i) dx, the pole is at z = i. Which contour is used? —
Upper semi-circle (pole is outside)
42.
For the integral ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> f(x)e^(iax) dx with a < 0, which contour and lemma are typically used? —
Lower semi-circle and Jordan's Lemma
43.
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: —
z² + 2z - 1 = 0
44.
The integral ∫₀<0xE2><0x88><0x9E> dx / (x² + 1) is evaluated using a semi-circular contour. Its value is: —
π/2
45.
What is the value of ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> dx / (x⁴ + 1)? —
π / √2
46.
What is the value of ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> x² / (x⁴ + 4) dx? —
π / √2
47.
Evaluate ∫₀²<0xC2><0x80><0x93> dθ / (2 + cos θ). —
2π / √3
48.
For the integral ∫₋<0xE2><0x88><0x9E>⁺<0xE2><0x88><0x9E> cos(x) / (x² + a²) dx (a > 0), the value is: —
πe⁻ᵃ / a
49.
The integral ∫₀<0xE2><0x88><0x9E> x^(m-1) / (1 + x) dx can be evaluated using a keyhole contour. The value is: —
π / sin(mπ)