8. Definite Integrals

75. Exploring powers of sine and cosine

c. Prove that ∫₀ᵖⁱ sin²(nx) dx has the same value for all positive integers n.

75. Exploring powers of sine and cosine

e. Repeat parts (a), (b), and (c) with sin²x replaced by sin⁴x. Comment on your observations.

Use Table 5.6 to evaluate the following definite integrals.                                                                                                                    

 (d) ∫₀^π/¹⁶ sec ² 4𝓍 d𝓍

{Use of Tech} Powers of sine and cosine It can be shown that

∫ from 0 to π/2 of sinⁿx dx = ∫ from 0 to π/2 of cosⁿx dx =

{

[1·3·5···(n-1)]/[2·4·6···n] · π/2 if n ≥ 2 is even

[2·4·6···(n-1)]/[3·5···n] if n ≥ 3 is odd

}

b. Evaluate the integrals with n = 10 and confirm the result.

Use geometry and properties of integrals to evaluate the following definite integrals.

∫₄⁰ (2𝓍 + √(16―𝓍²)) d𝓍 . (Hint: Write the integral as sum of two integrals.)

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

 ∫π/₁₆^π/⁸ 8 csc² 4𝓍 d𝓍

Definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result.

 ∫₀⁴ ƒ(𝓍) d𝓍, where ƒ(𝓍) = {5      if 𝓍 ≤ 2                                                                                                                                                                                     

                      3𝓍 ― 1  if 𝓍 > 2

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

 ∫π/₄^π/² (cos 𝓍) / (sin² 𝓍) d𝓍

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

(a) If ƒ is a constant function on the interval [a,b], then the right and left Riemann sums give the exact value of ∫ₐᵇ ƒ(𝓍) d𝓍, for any positive integer n.

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

(b) If ƒ is a linear function on the interval [a,b] , then a midpoint Riemann sums give the exact value of ∫ₐᵇ ƒ(𝓍) d𝓍, for any positive integer n. 

Use geometry and properties of integrals to evaluate

∫₀¹ (2𝓍 + √(1―𝓍²) + 1) d𝓍

Properties of integrals Suppose ∫₁⁴ ƒ(𝓍) d𝓍 = 6 , ∫₁⁴ g(𝓍) d𝓍 = 4 and ∫₃⁴ ƒ(𝓍) d𝓍 = 2 . Evaluate the following integrals or state that there is not enough information.

―∫₄¹ 2ƒ(𝓍) d𝓍

Properties of integrals Suppose ∫₁⁴ ƒ(𝓍) d𝓍 = 6 , ∫₁⁴ g(𝓍) d𝓍 = 4 and ∫₃⁴ ƒ(𝓍) d𝓍 = 2 . Evaluate the following integrals or state that there is not enough information.

∫₁³ ƒ(𝓍)/g(𝓍) d𝓍

Evaluating integrals Evaluate the following integrals.

∫₋₂² (3𝓍⁴―2𝓍 + 1) d𝓍

Evaluating integrals Evaluate the following integrals.

∫₀¹ √𝓍 (√𝓍 + 1) d𝓍