8. Definite Integrals

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

a. If m is a positive integer, then ∫[0 to π] cos^(2m+1)(x) dx = 0.

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

b. If m is a positive integer, then ∫[0 to π] sin^m(x) dx = 0.

Properties of integrals Use only the fact that ∫₀⁴ 3𝓍 (4 ―𝓍) d𝓍 = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.

(a) ∫₄⁰ 3𝓍(4 ― 𝓍) d(𝓍)

Properties of integrals Use only the fact that ∫₀⁴ 3𝓍 (4 ―𝓍) d𝓍 = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.

(b) ∫₀⁴ 𝓍(𝓍 ― 4) d(𝓍)

Properties of integrals Use only the fact that ∫₀⁴ 3𝓍 (4 ―𝓍) d𝓍 = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.

(c) ∫₄⁰ 6𝓍(4 ― 𝓍) d(𝓍)

Properties of integrals Use only the fact that ∫₀⁴ 3𝓍 (4 ―𝓍) d𝓍 = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.

(d) ∫₀⁸ 3𝓍(4 ― 𝓍) d(𝓍)

Properties of integrals Suppose ∫₀³ƒ(𝓍) d𝓍 = 2 , ∫₃⁶ƒ(𝓍) d𝓍 = ―5 , and ∫₃⁶g(𝓍) d𝓍 = 1. Evaluate the following integrals.

(a) ∫₀³ 5ƒ(𝓍) d𝓍

Properties of integrals Suppose ∫₀³ƒ(𝓍) d𝓍 = 2 , ∫₃⁶ƒ(𝓍) d𝓍 = ―5 , and ∫₃⁶g(𝓍) d𝓍 = 1. Evaluate the following integrals.

(b) ∫₃⁶ (―3g(𝓍)) d𝓍

Use symmetry to explain why.

∫⁴₋₄ (5𝓍⁴ + 3𝓍³ + 2𝓍² + 𝓍 + 1) d𝓍 = 2 ∫₀⁴ (5𝓍⁴ + 2𝓍² + 𝓍 + 1) d𝓍 .

Symmetry in integrals Use symmetry to evaluate the following integrals.

∫²⁰⁰₋₂₀₀ 2x⁵ dx

Symmetry in integrals Use symmetry to evaluate the following integrals.

∫²₋₂ (x² + x³) dx

Symmetry in integrals Use symmetry to evaluate the following integrals.

∫₋π/₂^π/² 5 sin θ dθ

Symmetry in integrals Use symmetry to evaluate the following integrals.

∫₋π/₄^π/⁴ sec² x dx

Symmetry in integrals Use symmetry to evaluate the following integrals.

∫²₋₂ [(x³ ― 4x) / (x² + 1)] 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.

(f) ∫ₐᵇ (2 ƒ(𝓍) ―3g (𝓍)) d𝓍 = 2 ∫ₐᵇ ƒ(𝓍) d𝓍 + 3 ∫₆ᵃ g(𝓍) d𝓍 .