7. Antiderivatives & Indefinite Integrals

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

 ∫ 8𝓍 cos (4𝓍² + 3) d𝓍, u = 4𝓍² + 3

Use Table 5.6 to evaluate the following indefinite integrals.                                                                                                               

 (d) ∫ cos 𝓍/7 d𝓍

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  

 ∫ 𝓍 csc 𝓍² cot 𝓍² d𝓍

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  

 ∫ sec 4w tan 4w dw

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  

 ∫ sec² (10𝓍 + 7) d𝓍

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  

 ∫ (sin⁵ 𝓍 + 3 sin³ 𝓍― sin 𝓍) cos 𝓍 d𝓍

Integrals with sin² 𝓍 and cos² 𝓍 Evaluate the following integrals.                                                                                                             

 ∫ sin² 𝓍 d𝓍

76-81. Table of integrals Use a table of integrals to evaluate the following integrals.

79. ∫ sec⁵x dx

Explain why or why not. Determine whether the following statements are true and give an explanation or counterexample.

d. ∫2 sin x cos x dx = −(1/2) cos 2x + C.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ, ƒ', and ƒ'' are continuous functions for all real numbers.                                                                                                                                                           

(c) ∫ sin 2𝓍 d𝓍 = 2 ∫ sin 𝓍 d𝓍 .

Evaluate the integrals in Exercises 39–56.

49. ∫3sec²t/(6 + 3tan(t)) dt

Evaluate the integrals in Exercises 39–56.

56. ∫sec(x)dx/√(ln(sec(x)+tan(x)))

Evaluate the integrals in Exercises 41–60.

41. ∫sinh(2x)dx

Evaluate the integrals in Exercises 41–60.

43. ∫6cosh(x/2 - ln3)dx

Evaluate the integrals in Exercises 41–60.

45. ∫tanh(x/7)dx