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 .

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.

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