Back7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.
31. ∫ √(x² - 8x) dx, x > 8
7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.
37. ∫ dx / √(x² + 10x), x >
71-74. Deriving formulas Evaluate the following integrals. Assume a and b are real numbers and n is a positive integer.
71. ∫[x/(ax + b)] dx (Hint: u = ax + b.)
Evaluate the following integrals.
∫ eˣ/(e²ˣ + 2eˣ + 17) dx
Evaluate the following integrals.
∫ x/(x² + 6x + 18) dx
6. Evaluate ∫ cos x √(100 − sin² x) dx using tables after performing the substitution u = sin x.
7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.
11. ∫ 3u / (2u + 7) du
7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.
15. ∫ x / √(4x + 1) dx
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
d. Using the substitution u = tan(x) in ∫ (tan²x / (tan x - 1)) dx leads to ∫ (u² / (u - 1)) du.
Explain why or why not. Determine whether the following statements are true and give an explanation or counterexample.
e. The best approach to evaluating ∫(x³ + 1)/(3x²) dx is to use the change of variables u = x³ + 1.
64. Using a computer algebra system, it was determined that
∫x(x+1)^8 dx = (x^10)/10 + (8x^9)/9 + (7x^8)/2 + 8x^7 + (35x^6)/3 + (56x^5)/5 + 7x^4 + (8x^3)/3 + x^2/2 + C.
Use integration by substitution to evaluate ∫x(x+1)^8 dx.
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ and ƒ' are continuous functions for all real numbers.
(g) ∫ ƒ' (g(𝓍))g' (𝓍) d(𝓍) = ƒ(g(𝓍)) + C .
Use Table 5.6 to evaluate the following indefinite integrals.
(a) ∫ e¹⁰ˣ d𝓍
Use a substitution of the form u = a𝓍 + b to evaluate the following indefinite integrals
∫(e³ˣ ⁺¹ d𝓍
Use a substitution of the form u = a𝓍 + b to evaluate the following indefinite integrals.
∫(𝓍 + 1)¹² d𝓍