• Review procedures for fitting an integrand to one of the basic integration rules.

Assignment

  • Vocabulary and teal boxes
  • p461 2, 3, 6, 8, 9, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44, 48 52–64 even, 72, 81–84, 94–98

Additional Resources

Fitting Integrands to Basic Integration Rules

This section is a collection of examples and strategies for integrating. Most you have seen before, but some of these are unique cases. I won’t copy them here, buy I will highlight the summary at the end.

Procedures for Fitting Integrands to Basic Integration Rules

Expand (numerator)

\[\begin{align} (1+e^x)^2 = 1 +2e^x +e^{2x} \end{align}\]

Separate numerator

\[\begin{align} \frac{1+x}{x^2+1} = \frac{1}{x^2+1} + \frac{x}{x^2+1} \end{align}\]

Complete the square

\[\begin{align} \frac{1}{\sqrt{2x-x^2}} = \frac{1}{\sqrt{1-(x-1)^2}} \end{align}\]

Divide improper rational functions

\[\begin{align} \frac{x^2}{x^2+1} = 1 - \frac{1}{x^2+1} \end{align}\]

Add and subtract terms in numerator

\[\begin{align} \frac{2x}{x^2+2x+1} &= \frac{2x+2-2}{x^2+2x+1} = \frac{2x+2}{x^2+2x+1} - \frac{2}{(x+1)^2} \end{align}\]

Use trigonometric identities

\[\begin{align} \cot^2 x = \csc^2 x - 1 \end{align}\]

Multiply and divide by Pythagorean conjugate

\[\begin{align} \frac{1}{1+\sin x} &= \left(\frac{1}{1 + \sin x}\right)\left(\frac{1 - \sin x}{1 - \sin x}\right) \\[1em] &= \frac{1-\sin x}{1-\sin^2 x} \\[1em] &= \frac{1-\sin x}{\cos^2 x} \\[1em] &= \sec^2 x - \frac{\sin x}{\cos^2 x} \end{align}\]