[Technology Exercise] 75. Find, to two decimal places, the x-coordinate of the centroid of the region in the first quadrant bounded by the x-axis, the curve y = arctan(x), and the line x = √3.

Volume: Find the volume generated by revolving one arch of the curve y = sin x about the x-axis.

Evaluate the integrals in Exercises 39–54.

∫ 1 / (cos θ + sin 2θ) dθ

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.

∫ from 0 to ∞ of (dθ / (θ² - 1))

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.

∫ e^(-3t) sin(4t) dt

Evaluate the integrals in Exercises 1–22.

∫₀^(π/6) 3cos⁵(3x) dx

Exercises 59–64 require the use of various trigonometric identities before you evaluate the integrals.

∫ sin(θ) sin(2θ) sin(3θ) dθ

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.

∫ (tan θ + 3 / sin θ) dθ

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.

∫₀¹ (4r dr) / √(1 − r⁴)

Evaluate the integrals in Exercises 33–52.

∫ from -π/4 to π/4 of 6 tan⁴(x) dx

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.

∫₋₁³ (4x² - 7) / (2x + 3) dx

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.

∫₁^∞ dx / x^1.001

Area: Find the area between the x-axis and the curve y = √(1 + cos 4x), for 0 ≤ x ≤ π.

Use any method to evaluate the integrals in Exercises 65–70.

∫ x cos³(x) dx

Evaluate the integrals in Exercises 1–22.

∫₀^π 8cos⁴(2πx) dx

Evaluate the integrals in Exercises 1–24 using integration by parts.

∫ θ cos(πθ) dθ

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.

∫₀^∞ 2e^(−θ) sinθ dθ

Find the value of the constant c so that the given function is a probability density function for a random variable X over the specified interval.

f(x) = c * x * √(25 - x²) over [0, 5]

Evaluate the integrals in Exercises 33–52.

∫ sec(x) tan²(x) dx

Evaluate the integrals in Exercises 33–52.

∫ cot³(t) csc⁴(t) dt

In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.

∫ dy / (y√(1 + (ln y)²)) from 1 to e

In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.

∫ dx / (x² + 2x)

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.

∫₂⁴ dt / [t√(t² − 4)]

Moment about y-axis:

A thin plate of constant density δ = 1 occupies the region enclosed by the curve

y = 36/(2x + 3) and the line x = 3 in the first quadrant. Find the moment of the plate about the y-axis.

In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.

∫₋∞⁴ [x / (x² + 9)^(2/5)] dx

In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.

∫ x^2 √(2x - x^2) dx

Evaluate the integrals in Exercises 23–32.

∫_{π/2}^{3π/4} √(1 - sin(2x)) dx

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.

∫ dx / (x √(7 - x²))

Use any method to evaluate the integrals in Exercises 55–66.

∫ (x + 2) / (x³ - 2x² - 3x) dx

Exercises 83–86 are about the infinite region in the first quadrant between the curve y = e^(-x) and the x-axis.

86. Find the volume of the solid generated by revolving the region about the x-axis.