Centroid: Find the centroid of the region bounded by the x-axis, the curve y = csc x, and the lines x = π/6, x = 5π/6.

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

∫ θ cos(πθ) dθ

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

∫ √(1 - (ln x)²) / (x ln x) dx

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 4 to ∞ of (dx / (√x - 1))

Volume: Find the volume of the solid generated by revolving the region in Exercise 45 about the x-axis.

Expand the quotients in Exercises 1–8 by partial fractions.

z / (z³ - z² - 6z)

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

83. Find the area of the region.