Evaluate the following improper integrals whenever they are convergent. ∫-∞ 0 e4x 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 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.
> If k > 0, show that ∫1 ∞ k / xk+1 dx = 1.
> If k > 0, show that ∫0 ∞ k e-kx dx = 1.
> Evaluate the following improper integrals whenever they are convergent. ∫-∞ ∞ e-x/(e-x + 2)2 dx
> Evaluate the following improper integrals whenever they are convergent. ∫0 ∞ e-x / (e-x + 2)2 dx
> Determine the integrals by making appropriate substitutions. ∫ x2/(3 - x3) dx
> Evaluate the following improper integrals whenever they are convergent. ∫-∞ 0 1/(24 – x) dx
> Evaluate the following improper integrals whenever they are convergent. ∫-∞ 0 6/(1 - 3x)2 dx
> Evaluate the following improper integrals whenever they are convergent. ∫-∞ 0 8/(x - 5)2 dx
> Evaluate the following improper integrals whenever they are convergent. ∫1 ∞ (5x + 1)-4 dx
> Evaluate the following improper integrals whenever they are convergent. ∫0 ∞ 2x(x2 + 1)-3/2 dx
> Evaluate the following improper integrals whenever they are convergent. ∫0 ∞ x/(x2 + 1) dx
> Evaluate the following improper integrals whenever they are convergent. ∫0 ∞ xe-x2 dx
> Evaluate the following improper integrals whenever they are convergent. ∫2 ∞ 1 / x ln x dx
> Evaluate the following improper integrals whenever they are convergent. ∫3 ∞ x2/ √(x3 – 1) dx
> Determine the integrals by making appropriate substitutions. ∫ ln√x / x dx
> Determine the integrals by making appropriate substitutions. ∫ 2(2x - 1)7 dx
> Evaluate the following improper integrals whenever they are convergent. ∫1 ∞ e-0.2x dx
> Evaluate the following improper integrals whenever they are convergent. ∫0 ∞ 6e1-3x dx
> Evaluate the following improper integrals whenever they are convergent. ∫0 ∞ 4/(2x + 1)3 dx
> Evaluate the following improper integrals whenever they are convergent. ∫0 ∞.01e- 0.01x dx
> Evaluate the following improper integrals whenever they are convergent. ∫2 ∞ e2-x dx
> Evaluate the following improper integrals whenever they are convergent. ∫2 ∞ 1/(x - 1)5/2 dx
> Evaluate the following improper integrals whenever they are convergent. ∫0 ∞ (x2 + 1) dx
> Evaluate the following improper integrals whenever they are convergent. ∫1 ∞ e2x dx
> Evaluate the following improper integrals whenever they are convergent. ∫0 ∞ e-3x dx
> Evaluate the following improper integrals whenever they are convergent. ∫0 1 1/ (2x + 3)2 dx
> Determine the integrals by making appropriate substitutions. ∫ x-2 (1/x + 2)5 dx
> Evaluate the following improper integrals whenever they are convergent. ∫1 ∞ 2/ x3/2 dx
> Evaluate the following improper integrals whenever they are convergent. ∫1 ∞ x3 dx
> Show that the region under the graph of y = (x - 1)-1/3 for x ≥ 2 cannot be assigned any finite number as its area.
> Show that the region under the graph of y = (14x + 18)-4/5 for x ≥ 1 cannot be assigned any finite number as its area. (See Fig. 9.) Figure 9: 16
> Find the area under the graph of y = (2x + 6)-4/3 for x ≥ 1. (See Fig. 8.) Figure 8: 1 16 Y
> Find the area under the graph of y = (x + 1)-3/2 for x ≥ 3.
> Find the area under the graph of y = 4 e-4x for x ≥ 0.
> Find the area under the graph of y = e-x/2 for x ≥ 0.
> Find the area under the graph of y = (x + 1)-2 for x ≥ 0.
> Find the area under the graph of y = 1/x2 for x ≥ 2.
> Determine the integrals by making appropriate substitutions. ∫ (x – 3)/(1 - 6x + x2)2 dx
> Determine if the given expression approaches a limit as b → ∞, and find that number when it does. 2 - e-3b
> Determine if the given expression approaches a limit as b → ∞, and find that number when it does. e-b/2 + 5
> Determine if the given expression approaches a limit as b → ∞, and find that number when it does. 4 (1 - b-3/4)
> Determine if the given expression approaches a limit as b → ∞, and find that number when it does. 5 (b2 + 3)-1
> Determine if the given expression approaches a limit as b → ∞, and find that number when it does. 5 - (b - 1)-1
> Find the present value of a continuous stream of income over 4 years if the rate of income is 25 e-0.02t thousand dollars per year at time t and the interest rate is 8%.
> Find the present value of a continuous stream of income over the time from t = 1 to t = 5 years when the interest rate is 10% and the income is produced at the rate of $12,000 per year.
> A continuous stream of income is being produced at the constant rate of $60,000 per year. Find the present value of the income generated during the time from t = 2 to t = 6 years, with a 6% interest rate.
> Find the present value of a continuous stream of income over 5 years when the rate of income is constant at $35,000 per year and the interest rate is 7%.
> Suppose that the population density function for a city is 40 e-0.5t thousand people per square mile. Let P(t) be the total population that lives within t miles of the city center, and let t be a small positive number. (a) Consider the ring about the cit
> Determine the following indefinite integrals: ∫ x2 ln x2 dx
> A volcano erupts and spreads lava in all directions. The density of the deposits at a distance t kilometers from the center is D (t) thousand tons per square kilometer, where D (t) = 11(t2 + 10)-2. Find the tonnage of lava deposited between the distances
> The population density of Philadelphia in 1940 was given by the function 60 e-0.4t. Calculate the number of people who lived within 5 miles of the city center. Sketch the graphs of the population densities for 1900 and 1940 (see Exercise 9) on a common g
> Use the population density from Exercise 9 to calculate the number of people who lived between 3 and 5 miles from the city center. Exercise 9: In 2012, the population density of a city t miles from the city center was 120 e-0.65t thousand people per squ
> In 2012, the population density of a city t miles from the city center was 120 e-0.65t thousand people per square mile. (a) Write a definite integral whose value equals the number of people (in thousands) who lived within 5 miles of the city center. (b)
> Find the present value of a stream of earnings generated over the next 2 years at the rate of 50 + 7t thousand dollars per year at time t assuming a 10% interest rate.
> A growth company is one whose net earnings tend to increase each year. Suppose that the net earnings of a company at time t are being generated at the rate of 30 + 5t million dollars per year. (a) Write a definite integral that gives the present value of
> A continuous stream of income is produced at the rate of 20 e1-0.09 t thousand dollars per year at time t, and invested money earns 6% interest. (a) Write a definite integral that gives the present value of this stream of income over the time from t = 2
> Find the present value of a continuous stream of income over 3 years if the rate of income is 80 e-0.08t thousand dollars per year at time t and the interest rate is 11%.
> Divide the given interval into n subintervals and list the value of Δx and the endpoints a0, a1, …. , an of the subintervals. -1 ≤ x ≤ 2; n = 5
> Divide the given interval into n subintervals and list the value of Δx and the endpoints a0, a1, …. , an of the subintervals. 3 ≤ x ≤ 5; n = 5
