Use the substitutions in Equations (1)–(4) to evaluate the integrals in Exercises 33–40. Integrals like these arise in calculating the average angular velocity of the output shaft of a universal joint when the input and output shafts are not aligned.
∫ dx / (1 + sin x + cos x)

2. When applying the formula for integration by parts, how do you choose the u and dv? How can you apply integration by parts to an integral of the form ∫ f(x) dx?

Finding surface area

Find the area of the surface generated by revolving the curve in Exercise 23 about the y-axis.

Finding volume

The infinite region bounded by the coordinate axes and the curve y = −ln x in the first quadrant is revolved about the x-axis to generate a solid. Find the volume of the solid.

Finding volume

The region in the first quadrant that is enclosed by the x-axis and the curve y = 3x√(1 − x) is revolved about the y-axis to generate a solid. Find the volume of the solid.

7. What is the goal of the method of partial fractions?

Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.

∫ [(x³ + 1) / (x³ − x)] dx