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3Bstudyguide - able to compute both types The following...

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This is a study guide for the second Math 3B midterm. It indi- cates which types of problems you may be expected to answer on the midterm, with instructions on where to find these topics in the Stewart calculus book. Plenty of examples can be found in the exercises at the end of each sec- tion of any calculus book. Although many of these may be a little more complicated than problems on the exam, they are excellent practice. In general, the best way to learn the techniques of integration is to practice a lot of problems. This document is intended to make some supplementary points. No books, notes, index cards, calculators of any kind are allowed. To the extent that you need trigonometric identities, these will be provided for you on the cover sheet. You are still responsible for knowing all the basic antiderivatives listed on the first page of the review sheet for Exam I (still posted on GauchoSpace. Remember that our techniques of integration apply to de- nite integrals (areas) as well as indenite integrals (antiderivatives).
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Unformatted text preview: able to compute both types. The following identities will be given to you on the exam. I might even put more if I’m feeling generous! • sin 2 x + cos 2 x = 1 • tan 2 x + 1 = sec 2 x • cos 2 x = 1 2 (1 + cos(2 x )) • sin 2 x = 1 2 (1-cos(2 x )) • sin(2 x ) = 2 sin( x ) cos( x ) Techniques you need to know: • u-substitution. For example, Z x ( x 2 + 1) 3 dx See Stewart 5.5 for more. • Integration by parts. For example, Z x sin( x ) dx See Stewart 7.1 for more. • Partial fractions. For example, Z x 2 + 3 x 3-2 x 2-3 x dx See Stewart 7.4 for more. • Inverse/trigonometric substitutions. For example, Z 1 x 2 √ x 2-16 dx See Stewart 7.3 for more. • Strategy for integrating i.e., identifying which of the above techniques to apply to a given integral and combining the above techniques to compute integrals. See Stewart 7.5. • You also need to know about improper integrals. For example Z ∞ 2 1 x 2 dx See Stewart 7.8 for more....
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