Without evaluating integrals, prove that ∫₀² d/dx(12 sin πx²) dx=∫₀² d/dx (x¹⁰(2−x)³) dx.

Area versus net area Graph the following functions. Then use geometry (not Riemann sums) to find the area and the net area of the region described.

The region between the graph of y = 1 - |x| and the x-axis, for -2 ≤ x ≤ 2

Evaluate the integrals in Exercises 39–56.

39. ∫(from -3 to -2)dx/x

Evaluate the integrals in Exercises 41–60.

51. ∫(from ln2 to ln4)coth(x)dx

Evaluate the integrals in Exercises 1–22.

∫₀^(π/6) 3cos⁵(3x) dx

Evaluating Definite Integrals

Evaluate the integrals in Exercises 47–68.

∫₋₁¹ (3x² - 4x + 7)dx

Evaluate the integrals in Exercises 47–68.

∫₁² 4 dv

v²

If ∫²₋₂ 3ƒ(x) dx = 12, ∫⁵₋₂ ƒ(x) dx = 6, and ∫⁵₋₂ g(x) dx = 2, find the value of each of the following.

a. ∫²₋₂ ƒ(x) dx