BackGaussians An important function in statistics is the Gaussian (or normal distribution, or bell-shaped curve), f(x) = e^(-ax²).
c. Complete the square to evaluate ∫ from -∞ to ∞ of e^(-(ax² + bx + c)) dx, where a > 0, b, and c are real numbers.
102–106. Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by F(s) = ∫[0 to ∞] e^(-st) f(t) dt, where we assume s is a positive real number. For example, to find the Laplace transform of f(t) = e^(-t), the following improper integral is evaluated using integration by parts:
F(s) = ∫[0 to ∞] e^(-st) e^(-t) dt = ∫[0 to ∞] e^(-(s+1)t) dt = 1/(s+1).
Verify the following Laplace transforms, where a is a real number.
106. f(t) = cos(at) → F(s) = s/(s² + a²)
95–98. {Use of Tech} Numerical methods Use numerical methods or a calculator to approximate the following integrals as closely as possible. The exact value of each integral is given.
96. ∫(from 0 to ∞) (sin²x)/x² dx = π/2
The Eiffel Tower Property Let R be the region between the curves y = e^(-c·x) and y = -e^(-c·x) on the interval [a, ∞), where a ≥ 0 and c > 0.
The center of mass of R is located at (x̄, 0), where x̄ = [∫(a to ∞) x·e^(-c·x) dx] / [∫(a to ∞) e^(-c·x) dx]
(The profile of the Eiffel Tower is modeled by these two exponential curves; see the Guided Project "The exponential Eiffel Tower")
d. Prove this property holds for any a ≥ 0 and c > 0:
The tangent lines to y = ±e^(-c·x) at x = a always intersect at R's center of mass
(Source: P. Weidman and I. Pinelis, Comptes Rendu, Mechanique, 332, 571-584, 2004)
Gamma function The gamma function is defined by Γ(p) = ∫ from 0 to ∞ of x^(p-1) e^(-x) dx, for p not equal to zero or a negative integer.
a. Use the reduction formula ∫ from 0 to ∞ of x^p e^(-x) dx = p ∫ from 0 to ∞ of x^(p-1) e^(-x) dx for p = 1, 2, 3, ...
to show that Γ(p + 1) = p! (p factorial).
Gamma function The gamma function is defined by Γ(p) = ∫ from 0 to ∞ of x^(p-1) e^(-x) dx, for p not equal to zero or a negative integer.
b. Use the substitution x = u² and the fact that ∫ from 0 to ∞ of e^(-u²) du = √(π/2) to show that Γ(1/2) = √π.
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₁^∞ dx / x^1.001
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₋₈¹ dx / x^(1/3)
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀¹ dr / r^0.999
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₋∞² (2 dx) / (x² + 4)
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₁^∞ dx / [x√(x² − 1)]
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀^∞ (16 tan⁻¹x dx) / (1 + x²)
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀^∞ 2e^(−θ) sinθ dθ
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₋∞^∞ 2x e^(−x²) dx
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₂⁴ dt / [t√(t² − 4)]