If ∫₀² ƒ(x) dx = π, ∫₀² 7g(x) dx = 7, and ∫₀¹ g(x) dx = 2, find the value of each of the following.
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d. ∫₀² √2ƒ(x) dx

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 1–22.

∫₀^(π/6) 3cos⁵(3x) dx

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

Evaluate the integrals in Exercises 23–32.

∫₀^π √(1 - cos²(θ)) dθ

Evaluate the integrals in Exercises 53–58.

∫ from 0 to π/2 of sin(x) cos(x) dx

Evaluate the integrals in Exercises 53–58.

∫ from -π/2 to π/2 of cos(x) cos(7x) dx

Evaluate the limits in Exercise 7 and 8.

lim (x → ∞) ∫₋ˣ^ˣ sin t dt