Evaluating integrals Evaluate the following integrals.


∫₋π/₂^π/² (cos 2𝓍 + cos 𝓍 sin 𝓍 ― 3 sin 𝓍⁵) 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.

(g) ∫ ƒ' (g(𝓍))g' (𝓍) d(𝓍) = ƒ(g(𝓍)) + C .

Evaluating integrals Evaluate the following integrals.                                                                                                                                         

 ∫ sin 𝒵 sin (cos 𝒵) d𝒵

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

 ∫₀⁴ √(8𝓍―𝓍²) d𝓍 . (Hint: Complete the square .)

Evaluating integrals Evaluate the following integrals.

∫₋₅⁵ ω³ /√(ω⁵⁰ + ω²⁰ + 1) dω (Hint: Use symmetry . )

Function defined by an integral Let H (𝓍) = ∫₀ˣ √(4 ― t²) dt, for ― 2 ≤ 𝓍 ≤ 2.

(c) Evaluate H '(2) .

Area functions and the Fundamental Theorem Consider the function

ƒ(t) = { t      if  ―2 ≤ t < 0

t²/2    if    0 ≤ t ≤ 2

and its graph shown below. Let F(𝓍) = ∫₋₁ˣ ƒ(t) dt and G(𝓍) = ∫₋₂ˣ ƒ(t) dt.

(c) Use the Fundamental Theorem to find an expression for F '(𝓍) for 0 ≤ 𝓍 < 2.