12. Techniques of Integration

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

∫ θ·cos(2θ + 1) dθ

Evaluate the integrals in Exercises 1–6.

∫ (arcsin x)² dx

Evaluate the integrals in Exercises 1–6.

∫ x arcsin x dx

Use the formula ∫ f⁻¹(x) dx = x f⁻¹(x) - ∫ f(y) dy, y = f⁻¹(x)

To evaluate the integrals in Exercises 77-80. Express your answers in terms of x.

∫ log₂ x dx

Finding volume

The region in the first quadrant enclosed by the coordinate axes, the curve y = e^x, and the line x = 1 is revolved about the y-axis to generate a solid. Find the volume of the solid.

143.

b. Find the average value of ln(x) over [1, e].

18. Finding volume (Continuation of Exercise 17.) Find the volume of the solid generated by revolving the region R about:

a. the y-axis.

Finding volume: Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes and the curve y = cos(x), 0 ≤ x ≤ π/2, about

b. The line x = π/2.

Finding volume: Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve y = e^x, and the line x = ln(2) about the line x = ln(2).

Finding volume: Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve y = e^(-x), and the line x = 1.

a. About the y-axis.

Finding volume: Find the volume of the solid generated by revolving the region bounded by the x-axis and the curve y = x sin(x), 0 ≤ x ≤ π, about

a. The y-axis.

(See Exercise 57 for a graph.)

Consider the region bounded by the graphs of

y = arctan(x), y = 0, and x = 1.

b. Find the volume of the solid formed by revolving this region about the y-axis.

Use integration by parts to obtain the formula ∫ √(1 - x²) dx = (1/2) x √(1 - x²) + (1/2) ∫ 1 / √(1 - x²) dx.

Equations (4) and (5) lead to different formulas for the integral of arctan x:

a. ∫ arctan x dx = x arctan x - ln sec(arctan x) + C [Eq. (4)]

b. ∫ arctan x dx = x arctan x - ln √(1 + x²) + C [Eq. (5)]

Can both integrations be correct? Explain.

19. Center of mass Find the center of mass of a thin plate of constant density covering the region in the first and fourth quadrants enclosed by the curves y=1/(1+x²) and y=-1/(1+x²) and by the lines x=0 and x=1.