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07:33{Use of Tech} Using the integral of sec³u By reduction formula 4 in Section 8.3,
∫sec³u du = 1/2 (sec u tan u + ln |sec u + tan u|) + C
Graph the following functions and find the area under the curve on the given interval.
f(x) = (9 - x²) ⁻², [0, 3/2]
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)
87-92. An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational function using the substitution u = tan(x/2) or, equivalently, x = 2 tan⁻¹u. The following relations are used in making this change of variables.
A: dx = 2/(1 + u²) du
B: sin x = 2u/(1 + u²)
C: cos x = (1 - u²)/(1 + u²)
88. Evaluate ∫ dx/(2 + cos x).
9–61. Trigonometric integrals Evaluate the following integrals.
38. ∫ tan⁵θ sec⁴θ dθ
79–82. {Use of Tech} Double table look-up The following integrals may require more than one table look-up. Evaluate the integrals using a table of integrals, and then check your answer with a computer algebra system.
82. ∫ (sin⁻¹(ax)) / x² dx, a > 0
7–58. Improper integrals Evaluate the following integrals or state that they diverge.
42. ∫ (from 3 to 4) 1/(x-3)³ᐟ² dx
