BackDefinite Integrals
In Exercises 5–8, express each limit as a definite integral. Then evaluate the integral to find the value of the limit. In each case, P is a partition of the given interval, and the numbers cₖ are chosen from the subintervals of P.
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lim ∑ (2cₖ - 1)⁻¹/² ∆xₖ, where P is a partition of [1, 5]
∥P∥→0 k = 1
Definite Integrals
In Exercises 5–8, express each limit as a definite integral. Then evaluate the integral to find the value of the limit. In each case, P is a partition of the given interval, and the numbers cₖ are chosen from the subintervals of P.
n
lim ∑ (cos(cₖ/2)) ∆xₖ, where P is a partition of [-π, 0]
∥P∥→0 k = 1
Evaluate the integrals in Exercises 1–22.
∫₀^(π/2) sin²(2θ) cos³(2θ) dθ
Evaluate the integrals in Exercises 23–32.
∫₀^(π/6) √(1 + sin(x)) dx
(Hint: Multiply by √((1 - sin(x)) / (1 - sin(x))))
Evaluate the integrals in Exercises 23–32.
∫₋π^π (1 - cos²(t))^(3/2) dt
Evaluate the integrals in Exercises 33–52.
∫ from -π/4 to π/4 of 6 tan⁴(x) dx
Use the substitutions in Equations (1)–(4) to evaluate the integrals in Exercises 33–40. Integrals like these arise in calculating the average angular velocity of the output shaft of a universal joint when the input and output shafts are not aligned.
∫(from π/2 to 2π/3) cos θ dθ / (sin θ cos θ + sin θ)
Evaluate the integrals in Exercises 41–60.
55. ∫(from -π/4 to π/4)cosh(tanθ)sec²θ dθ
Evaluate the integrals in Exercises 41–60.
57. ∫(from 1 to 2)cosh(ln t)/t dt
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫₀^(π/3) tan³x·sec²x dx
Evaluate the integrals in Exercises 1–22.
∫₀^π 8cos⁴(2πx) dx