11. Integrals of Inverse, Exponential, & Logarithmic Functions

76. Apparent discrepancy

Three different computer algebra systems give the following results:

∫ (dx / (x√(x⁴ − 1))) = ½ cos⁻¹(√(x⁻⁴)) = ½ cos⁻¹(x⁻²) = ½ tan⁻¹(√(x⁴ − 1)).

Explain how all three can be correct.

Use Table 5.6 to evaluate the following definite integrals.                                                                                                                    

 (c) ∫₃√₂^⁶ d𝓍/(𝓍² ―9)

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

 ∫ 2 / (𝓍√4𝓍² ―1) d𝓍 , 𝓍 > ½ 

Evaluating integrals Evaluate the following integrals.

∫√₂/₅^²/⁵ d𝓍/𝓍√(25𝓍² ―1)

60–69. Completing the square Evaluate the following integrals.

65. ∫[1/2 to (√2 + 3)/(2√2)] dx / (8x² - 8x + 11)

Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.                                                                                                                         

 ∫₂/₍₅√₃₎^²/⁵ d𝓍/ x√(25𝓍²― 1)

37–56. Integrals Evaluate each integral.

∫ cosh 2x dx

37–56. Integrals Evaluate each integral.

∫₀ ˡⁿ ² tanh x dx

37–56. Integrals Evaluate each integral.

∫ sinh x / (1 + cosh x) dx

37–56. Integrals Evaluate each integral.

∫ tanh²x dx (Hint: Use an identity.)

37–56. Integrals Evaluate each integral.

∫₀⁴ sech²√x / √x dx

37–56. Integrals Evaluate each integral.

∫ sinh²z dz (Hint: Use an identity.)

37–56. Integrals Evaluate each integral.

∫ sech² w tanh w dw

37–56. Integrals Evaluate each integral.

∫ (cosh z) / (sinh² z) dz

57–58. Two ways

Evaluate the following integrals two ways.

a. Simplify the integrand first and then integrate.

b. Change variables (let u = ln x), integrate, and then simplify your answer. Verify that both methods give the same answer.

∫ (sinh (ln x)) / x dx