12. Techniques of Integration

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

a. the y-axis.

For the integral ${\int }_{}^{}x{e}^{x}dx$, which of the following correctly identifies $u$ and $du$ for use in integration by parts?

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

38. ∫ (from π/4 to π/2) x csc²x dx

9–40. Integration by parts Evaluate the following integrals using integration by parts.

11. ∫ t · e⁶ᵗ dt

81. Possible and impossible integrals

Let Iₙ = ∫ xⁿ e⁻ˣ² dx, where n is a nonnegative integer.

d. Show that, in general, if n is odd, then Iₙ = -½ e⁻ˣ² pₙ₋₁(x), where pₙ₋₁ is a polynomial of degree n - 1.

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

143.

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

9–40. Integration by parts Evaluate the following integrals using integration by parts.

14. ∫ s · e⁻²ˢ ds

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

60. ∫ x² coshx dx

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.

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

51. ∫ x²/√(4 + x²) dx

9–40. Integration by parts Evaluate the following integrals using integration by parts.

26. ∫ t³ sin(t) dt

58. Two Methods Evaluate ∫(from 0 to π/3) sin(x) · ln(cos(x)) dx in the following two ways:

b. Use substitution.

79. Tabular integration extended Refer to Exercise 77.

a. The following table shows the method of tabular integration applied to

∫ eˣ cos x dx.

Use the table to express ∫ eˣ cos x dx in terms of the sum of functions and an indefinite integral.

b. Solve the equation in part (a) for ∫ eʳ cos z dz.

c. Evaluate ∫ e⁻ᶻ sin 3z dz by applying the idea from parts (a) and (b).

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

∫x e^(3x) dx