MATA35H3 Study Guide - Final Guide: Trigonometric Substitution, Improper Integral
LECTURE SUMMARY 2.2
FRIDAY, MAY 13, 2016
Part I
1. Differences and similarities between type I and type II Improper integral.
2. lim
b→+∞
b1−p
1−p.
3. why R1
0
1
xdx diverges, while R1
0
1
√xdx converges.
4. Definition of Average Value of a continuous function.
5. Table of Trigonometric substitution.
Table of Trigonometric Substitution
Expression Substitution Identity
√a2−x2x=asin θ,−π
2≤θ≤π
21−sin2θ= cos2θ
√a2+x2x=atan θ,−π
2≤θ≤π
21 + tan2θ= sec2θ
√x2−a2x=asec θ,0 ≤θ < π
2or π≤θ < 3π
2sec2θ−1 = tan2θ
6. Key for trigonometric substitution: get rid of square root via substitution.
Part 2: Integration of Rational Functions by Partial Fractions
1. Definition of Rational Function.
2. Mathematical view of generalization.
3. R1
x−adx, for any constant a.
4. R1
x(x−1) dx.
5. For more examples, see problem set 2.2.
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