12. Techniques of Integration

62. Two integration methods Evaluate ∫ sin x cos x dx using integration by parts. Then evaluate the integral using a substitution. Reconcile your answers

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

17. ∫ x · 3x dx

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

28. ∫ ln² x dx

54-57. Applying Reduction Formulas Use the reduction formulas from Exercises 50-53 to evaluate the following integrals:

55. ∫ x² cos(5x) dx

Evaluate the definite integral.

${\int }_0^{\pi }{t}^2\sin tdt$

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

32. ∫ from 0 to 1 x² 2ˣ dx

Use a substitution to reduce the following integrals to ∫ ln u du. Then evaluate using the formula for ∫ ln x dx.

7. ∫ (sec²x) · ln(tan x + 2) dx

Evaluate the indefinite integral.

${\int }_{}^{}{x}^2\cos 4xdx$

Find the integral.

${\int }_{}^{}s{e}^{3s}ds$

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

∫ θ cos(πθ) dθ

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.

7–84. Evaluate the following integrals.

77. ∫ arccosx dx

[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.

71-74. Deriving formulas Evaluate the following integrals. Assume a and b are real numbers and n is a positive integer.

74. ∫xⁿ arcsin(x) dx (Hint: integration by parts.)

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.

∫ arctan x dx