Describe integration by substitution in your own words.
> 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. ∫
> 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.
> 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. ∫-∞ 0 e4x 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.
