8. Definite Integrals

Evaluate the integrals in Exercises 1–22.

∫₀^π 8 sin⁴(y) cos²(y) dy

Evaluate the integrals in Exercises 47–68.

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∫₁⁴ (1 + √u)¹/² du

√u

Evaluate the integrals in Exercises 47–68.

∫₀¹ dr .

∛(7 - 5r)²

Evaluate the integrals in Exercises 47–68.

∫₀¹/² x³ (1 + 9x⁴)⁻³/² dx

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.

∫ dx / (4 - x²)^(3/2) from 0 to 1

Evaluate the integrals in Exercises 47–68.

∫₀ ^π tan² (θ/3) dθ

Evaluate the integrals in Exercises 47–68.

∫₀ ^π/2 5(sin x)³/² cos x dx

Evaluate the integrals in Exercises 47–68.

∫₀^π/4 sec²x / (1 + 7 tan x)²/³ dx

Definite Integrals

In Exercises 5–8, express each limit as a definite integral. Then evaluate the integral to find the value of the limit. In each case, P is a partition of the given interval, and the numbers cₖ are chosen from the subintervals of P.

     n

 lim  ∑ (2cₖ - 1)⁻¹/² ∆xₖ, where P is a partition of [1, 5]

 ∥P∥→0  k = 1

Definite Integrals

In Exercises 5–8, express each limit as a definite integral. Then evaluate the integral to find the value of the limit. In each case, P is a partition of the given interval, and the numbers cₖ are chosen from the subintervals of P.

     n

 lim  ∑ (cos(cₖ/2)) ∆xₖ, where P is a partition of [-π, 0]

 ∥P∥→0  k = 1

Evaluate the integrals in Exercises 1–22.

∫₀^(π/2) sin²(2θ) cos³(2θ) dθ

Evaluate the integrals in Exercises 23–32.

∫₀^(π/6) √(1 + sin(x)) dx

(Hint: Multiply by √((1 - sin(x)) / (1 - sin(x))))

Evaluate the integrals in Exercises 23–32.

∫_{π/2}^{3π/4} √(1 - sin(2x)) dx

Evaluate the integrals in Exercises 23–32.

∫₋π^π (1 - cos²(t))^(3/2) dt

Evaluate the integrals in Exercises 33–52.

∫ from -π/4 to π/4 of 6 tan⁴(x) dx