7. Antiderivatives & Indefinite Integrals

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

Use the given substitution to evaluate the following indefinite integrals. Check your answer by differentiating.                                                                                              

 ∫ (6𝓍 + 1) √(3𝓍² + 𝓍) d𝓍 , u = 3𝓍² + 𝓍

Use Table 5.6 to evaluate the following indefinite integrals.                                                                                                               

 (f) ∫ d𝓍/√36 ―𝓍²

Use Table 5.6 to evaluate the following indefinite integrals.                                                                                                               

 (e) ∫ d𝓍/(81 + 9𝓍²) (Hint: Factor a 9 out of the denominator first.)