SINGLE VARIABLE CALCULUS II
SINGLE VARIABLE CALCULUS II MATH 102
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This 2 page Class Notes was uploaded by Jayde Lang on Monday October 19, 2015. The Class Notes belongs to MATH 102 at Rice University taught by Staff in Fall. Since its upload, it has received 24 views. For similar materials see /class/224930/math-102-rice-university in Mathematics (M) at Rice University.
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Date Created: 10/19/15
Math 102 001 Spring 2008 Stolz Some recommendations on using integration techniques 1 Substitution o A substitution u g works best if the derivative g appears as part of the integrand but there are exceptions o A substitution can be used to use an integrand to a form which can be treated with other methods eg one of the forms suitable for trigonometric substitution Example z5 3 7 1dz 2 Integration by Parts If an integrand has two factors which of them should one integrate and which one differentiate when using integration by part As a rule First choose log functions to differentiate If no log functions are present choose power functions Examples ln zew dx choose u m d1 emdz while in z lnmdz choose u lnx d1 mdx it would not be good to choose u m d1 ln mdz here In integrals of the form ew sinzdz choose u sinx and d1 emdz do two integrations by part and then solve for the unknown integral 0 Example zem sinzdm Start by nding few sinzdz and then choose u x d1 em sinzdz to integrate the original integral by parts 3 Trigonometric lntegrals o lntegrals of the form f sinm zcosnzdz If m or n are odd then use sinmcos2x 1 on one of the odd powered terms and then substitute u sinx or u cos x Example fsin5xcos4 dx f1 7 cos2 x2sincos3zdz Substitute u cos m If m and n are both odd then reduce the integrand to odd powered sines or cosines by using half angle identities o Integrals of the form ftanm xsecnzdz If m is odd use the identity tanzx seczz 7 1 to replace all but one of the tangent factors by secants then substitute u sec x If n is even use sec2 z 1 717 tan2 and substitute u tan x Example ftan3 ssecgzdz fsec2 z 7 1 sec2 secstanz dx fu4 7 u du 4 Partial Fractions o If necessary start by long division to turn an improper rational function into a proper rational function 0 Factor the denominator If the denominator can be fully factored use a standard77 partial fraction decomposition If the denominator contains ir reducible quadratic factors then other methods have to be used completing the square substitution to reduce to simpler forms of the denominator eg 2 1 splitting numerator 5 Trigonometric Substitutions o All three types of trigonometric substitutions are based on trying to exploit 1 7 sin2 m cos2 x and its two consequences tan2 z sec2 z 7 1 and sec2 m 1 tan2 x These identities directly suggest the proper substitutions to simplify expressions involving the terms a2 7 2 a2 x2 or 2 7 a2 0 Example 2x 7 x2 dz 6 lmproper lntegrals 0 Start by splitting an improper integral by splitting it at all singularities meaning at 00 700 and all discontinuities of the integrand Treat each term separately as an improper integral 0 When evaluating an improper integral rst nd the inde nite integral in a calculation on the side Then insert the antiderivative found into the correct boundaries for the improper integral Do not try to directly substitute integration boundaries in an improper integral 0 Example0 Wdz