12. Techniques of Integration

Finding volume

Let R be the "triangular" region in the first quadrant that is bounded above by the line y = 1, below by the curve y = ln x, and on the left by the line x = 1.

Find the volume of the solid generated by revolving R about

a. the x-axis.

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?

Find the integral.

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

Find the integral.

${\int }_{}^{}\theta \cos 2\theta d\theta$

Find the integral.

${\int }_{}^{}{x}^2\ln xdx$

Find the indefinite integral.

${\int }_{}^{}{r}^2{e}^{-r}dr$

Find the indefinite integral.

${\int }_{}^{}{\theta }^2\cos 3\theta d\theta$

Evaluate the indefinite integral.

${\int }_{}^{}(s^2+4)e^{3s}ds$

Evaluate the indefinite integral.

${\int }^{}\frac{\ln x}{x^4}dx$

Evaluate the indefinite integral.

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

Evaluate the definite integral.

${\int }_1^{2}x\ln xdx$

Evaluate the definite integral.

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

92–98. Evaluate the following integrals.

92. ∫[1 to √2] y⁸ e^(y²) dy

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

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.)