Decide whether integration by parts or a substitution should be used to compute the indefinite integral. If substitution, indicate the substitution to be made. If by parts, indicate the functions f (x) and g(x) to be used in formula (1) of Section 9.2. ∫ (x + 1)2 e3x dx
> Determine the following integrals using the indicated substitution. ∫ x sec2 x2 dx; u = x2
> Determine the following integrals using the indicated substitution. ∫ x4 / (x5 – 7) ln(x5 - 7) dx; u = ln(x5 - 7)
> Determine the following integrals using the indicated substitution. ∫ (x + 5)-1/2 e√(x+5) dx; u = √(x+5)
> Determine the integrals by making appropriate substitutions. ∫ (x2 + 2x + 3)6(x + 1) dx
> Figure 2 shows graphs of several functions f (x) whose slope at each x is (2√x + 1)/ √x. Find the expression for the function f (x) whose graph passes through (4, 15). Figure 2: 15 10 5 0 Y 2 (4, 15) 4
> Figure 1 shows graphs of several functions f (x) whose slope at each x is x/√(x2 + 9). Find the expression for the function f (x) whose graph passes through (4, 8). Figure 1: 1 16+ 0 (4,8) 2 4 6 8
> Determine the integral by making appropriate substitutions. ∫ (e2x - 1) / (e2x + 1) dx
> Determine the integral by making appropriate substitutions. ∫ 1 / (1 + ex) dx
> Determine the integral by making appropriate substitutions. ∫ (1 + e-x)3 / ex dx
> Determine the integral by making appropriate substitutions. ∫ (ex + e-x) / (ex - e-x) dx
> Determine the integral by making appropriate substitutions. ∫ (ex + e-x) / (ex - e-x) dx
> Determine the integral by making appropriate substitutions. ∫ ex / (1 + 2ex) dx
> Determine the integral by making appropriate substitutions. ∫ ex √(1 + ex) dx
> Determine the integrals by making appropriate substitutions. ∫ ex (1 + ex)5 dx
> Determine the integrals by making appropriate substitutions. ∫ (2x + 1)/ √(x2 + x + 3) dx
> Determine the following indefinite integrals: ∫ x(1 - 3x2)5 dx
> Determine the following indefinite integrals: ∫ √(2x + 1) dx
> Determine the integrals by making appropriate substitutions. ∫ dx / (3 - 5x)
> Determine the following indefinite integrals: ∫ x sin 3x2 dx
> The capitalized cost of an asset is the total of the original cost and the present value of all future “renewals” or replacements. This concept is useful, for example, when you are selecting equipment that is manufactured by several different companies.
> Suppose that a machine requires daily maintenance, and let M(t) be the annual rate of maintenance expense at time t. Suppose that the interval 0 ≤ t ≤ 2 is divided into n subintervals, with endpoints t0 = 0, t1, … , tn = 2. (a) Give a Riemann sum that ap
> Suppose that t miles from the center of a certain city the property tax revenue is approximately R(t) thousand dollars per square mile, where R(t) = 50 e-t/20. Use this model to predict the total property tax revenue that will be generated by property wi
> Find the present value of a continuous stream of income over the next 4 years, where the rate of income is 50e-0.08t thousand dollars per year at time t, and the interest rate is 12%.
> Let k be a positive number. It can be shown that lim b→∞ b e-kb = 0. Use this fact to compute ∫0 ∞ x e-kx dx.
> It can be shown that lim b→∞ b e-b = 0. Use this fact to compute ∫1 ∞ x e-x dx.
> Evaluate the following improper integrals whenever they are convergent. ∫-∞ 0 8/(5 - 2x)3 dx
> Evaluate the following improper integrals whenever they are convergent. ∫-1 ∞ (x + 3)-5/4 dx
> Evaluate the following improper integrals whenever they are convergent. ∫1 ∞ x2 e-x3 dx
> Determine the integrals by making appropriate substitutions. ∫ (3 - x) (x2 - 6x)4 dx
> Evaluate the following improper integrals whenever they are convergent. ∫1 ∞ (x + 2)/(x2 + 4x – 2) dx
> Evaluate the following improper integrals whenever they are convergent. ∫1 ∞ x-2/3 dx
> Evaluate the following improper integrals whenever they are convergent. ∫0 ∞ e6-3x dx.
> Approximate the following definite integrals by the midpoint rule, the trapezoidal rule, and Simpson’s rule. ∫-1 1 1/(1 + x2) dx; n = 5
> Approximate the following definite integrals by the midpoint rule, the trapezoidal rule, and Simpson’s rule. ∫1 4 ex / (x + 1) dx; n = 5
> Approximate the following definite integrals by the midpoint rule, the trapezoidal rule, and Simpson’s rule. ∫0 10 e√x dx; n = 5
> Approximate the following definite integrals by the midpoint rule, the trapezoidal rule, and Simpson’s rule. ∫1 9 1/√x dx; n = 4
> Evaluate the following definite integrals: ∫1 2x-3/2 ln x dx
> Evaluate the following definite integrals: ∫1 2 x e-2x dx
> Evaluate the following definite integrals: ∫1/2 1 ln(2x + 3) / (2x + 3) dx
> Determine the integrals by making appropriate substitutions. ∫ 2/ x (ln x)4 dx
> Evaluate the following definite integrals: ∫0 2 x e-(1/2) x2 dx
> Evaluate the following definite integrals: ∫0 π/2 x sin 8x dx
> Evaluate the following definite integrals: ∫0 1 2x / (x2 + 1)3 dx
> Decide whether integration by parts or a substitution should be used to compute the indefinite integral. If substitution, indicate the substitution to be made. If by parts, indicate the functions f (x) and g(x) to be used in formula (1) of Section 9.2. ∫
> Decide whether integration by parts or a substitution should be used to compute the indefinite integral. If substitution, indicate the substitution to be made. If by parts, indicate the functions f (x) and g(x) to be used in formula (1) of Section 9.2. ∫
> Decide whether integration by parts or a substitution should be used to compute the indefinite integral. If substitution, indicate the substitution to be made. If by parts, indicate the functions f (x) and g(x) to be used in formula (1) of Section 9.2. ∫
> Decide whether integration by parts or a substitution should be used to compute the indefinite integral. If substitution, indicate the substitution to be made. If by parts, indicate the functions f (x) and g(x) to be used in formula (1) of Section 9.2. ∫
> Decide whether integration by parts or a substitution should be used to compute the indefinite integral. If substitution, indicate the substitution to be made. If by parts, indicate the functions f (x) and g(x) to be used in formula (1) of Section 9.2. ∫
> Decide whether integration by parts or a substitution should be used to compute the indefinite integral. If substitution, indicate the substitution to be made. If by parts, indicate the functions f (x) and g(x) to be used in formula (1) of Section 9.2. ∫
> Determine the integrals by making appropriate substitutions. ∫ 1 / x ln x2
> Decide whether integration by parts or a substitution should be used to compute the indefinite integral. If substitution, indicate the substitution to be made. If by parts, indicate the functions f (x) and g(x) to be used in formula (1) of Section 9.2. ∫
> Decide whether integration by parts or a substitution should be used to compute the indefinite integral. If substitution, indicate the substitution to be made. If by parts, indicate the functions f (x) and g(x) to be used in formula (1) of Section 9.2. ∫
> Decide whether integration by parts or a substitution should be used to compute the indefinite integral. If substitution, indicate the substitution to be made. If by parts, indicate the functions f (x) and g(x) to be used in formula (1) of Section 9.2. ∫
> Decide whether integration by parts or a substitution should be used to compute the indefinite integral. If substitution, indicate the substitution to be made. If by parts, indicate the functions f (x) and g(x) to be used in formula (1) of Section 9.2. ∫
> Decide whether integration by parts or a substitution should be used to compute the indefinite integral. If substitution, indicate the substitution to be made. If by parts, indicate the functions f (x) and g(x) to be used in formula (1) of Section 9.2. ∫
> Decide whether integration by parts or a substitution should be used to compute the indefinite integral. If substitution, indicate the substitution to be made. If by parts, indicate the functions f (x) and g(x) to be used in formula (1) of Section 9.2. ∫
> Decide whether integration by parts or a substitution should be used to compute the indefinite integral. If substitution, indicate the substitution to be made. If by parts, indicate the functions f (x) and g(x) to be used in formula (1) of Section 9.2. ∫
> Decide whether integration by parts or a substitution should be used to compute the indefinite integral. If substitution, indicate the substitution to be made. If by parts, indicate the functions f (x) and g(x) to be used in formula (1) of Section 9.2. ∫
> Decide whether integration by parts or a substitution should be used to compute the indefinite integral. If substitution, indicate the substitution to be made. If by parts, indicate the functions f (x) and g(x) to be used in formula (1) of Section 9.2. ∫
> Decide whether integration by parts or a substitution should be used to compute the indefinite integral. If substitution, indicate the substitution to be made. If by parts, indicate the functions f (x) and g(x) to be used in formula (1) of Section 9.2. ∫
> Determine the integrals by making appropriate substitutions. ∫ 3 / (2x + 1)3 dx
> Decide whether integration by parts or a substitution should be used to compute the indefinite integral. If substitution, indicate the substitution to be made. If by parts, indicate the functions f (x) and g(x) to be used in formula (1) of Section 9.2. ∫
> Determine the following indefinite integrals: ∫ x (ln x)2 dx
> Determine the following indefinite integrals: ∫ x / (1 – x)5 dx
> Determine the following indefinite integrals: ∫ x √(3x - 1) dx
> Determine the following indefinite integrals: ∫ x √(x + 1) dx
> Determine the following indefinite integrals: ∫ ln x2 dx
> Determine the following indefinite integrals: ∫ ln(ln x) / x ln x dx
> Determine the following indefinite integrals: ∫ x2 cos 3x dx
> Determine the following indefinite integrals: ∫ x ln(x2 + 1) /( x2 + 1) dx
> Determine the following indefinite integrals: ∫ x2 e-x3 dx
> Determine the integrals by making appropriate substitutions. ∫ 8x/ ex2 dx
> Determine the following indefinite integrals: ∫ x sin 3x dx
> Determine the following indefinite integrals: ∫ x √(4 - x2) dx
> Determine the following indefinite integrals: ∫ 1/√(4x + 3) dx
> Determine the following indefinite integrals: ∫ (ln x)2/x dx
> Determine the following indefinite integrals: ∫ (ln x)5/x dx
> Describe the change of limits rule for the integration by substitution of a definite integral.
> Describe integration by parts in your own words.
> Describe integration by substitution in your own words.
> How do you determine whether an improper integral is convergent?
> State the formula for each of the following quantities: (a) Present value of a continuous stream of income (b) Total population in a ring around the center of a city
> Determine the integrals by making appropriate substitutions. ∫ ln(3x) /3x dx
> State the error of approximation theorem for each of the three approximation rules.
> Explain the formula S = (2M + T)/3.
> State the trapezoidal rule. (Include the meaning of all symbols used.)
> State the midpoint rule. (Include the meaning of all symbols used.)
> State the formula for the integration by parts of a definite integral.
> Determine if the given expression approaches a limit as b → ∞, and find that number when it does. 2 - (b + 1)-1/2
> Determine if the given expression approaches a limit as b → ∞, and find that number when it does. ½ √b
> Determine if the given expression approaches a limit as b → ∞, and find that number when it does. ¼ - 1/b2
> Determine if the given expression approaches a limit as b → ∞, and find that number when it does. 1/b + 1/3
> Determine if the given expression approaches a limit as b → ∞, and find that number when it does. -3 e2b
> Determine the integrals by making appropriate substitutions. ∫ (x2 - 2x) / (x3 - 3x2 + 1)
> Determine if the given expression approaches a limit as b → ∞, and find that number when it does. b2
> Determine if the given expression approaches a limit as b → ∞, and find that number when it does. 5/b
> The capital value of an asset such as a machine is sometimes defined as the present value of all future net earnings. (See Section 9.5.) The actual lifetime of the asset may not be known, and since some assets may last indefinitely, the capital value of
> The capital value of an asset such as a machine is sometimes defined as the present value of all future net earnings. (See Section 9.5.) The actual lifetime of the asset may not be known, and since some assets may last indefinitely, the capital value of
> The capital value of an asset such as a machine is sometimes defined as the present value of all future net earnings. (See Section 9.5.) The actual lifetime of the asset may not be known, and since some assets may last indefinitely, the capital value of
> If k > 0, show that ∫e ∞ k / x(ln x)k+1 dx = 1.
