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Notes for Week 5

by: Bethany Lawler

Notes for Week 5 Math 182

Bethany Lawler
GPA 3.96
Honors Calculus II
S. Lapin

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About this Document

This covers trigonometric integration techniques, trigonometric substitutions, and partial fractions.
Honors Calculus II
S. Lapin
Class Notes
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This 2 page Class Notes was uploaded by Bethany Lawler on Friday September 25, 2015. The Class Notes belongs to Math 182 at Washington State University taught by S. Lapin in Summer 2015. Since its upload, it has received 38 views. For similar materials see Honors Calculus II in Math at Washington State University.


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Date Created: 09/25/15
Math 182 Notes Week 5 921 925 Integration by Parts 0 Integration by parts splits integrands that are comprised of two function multiplied by each other that cannot be solved by u substitution 0 The general formula for integration by parts is judvuv jvdu 0 u is one of the functions make up the integrand and dv is the other du is found by deriving u and v is found by integrating dv Trigonometric Integrands 0 There are two ways of finding the integral of single trigonometric function based on whether they are of odd or even power 0 If the power is odd then it is split into the an even power multiplied by a first power ex sin3x goes to sin2XsinX o The even power is then converted by either of the following formulas into its corresponding function 3139an coszx 1 tanzx 1 seczx 0 Then use u substitution on the new function this will eliminate the extra function leftover from the beginning 0 The rest is a matter of distributing and integrating o If the power is even then for sine and cosine the double angle formula can be used to decrease the power The double angle can be used as many times as necessary provided 1 6052x 1t 1s an even power the double angle formulas are smzx and coszx 1c052x 2 o For tangent and secant a reduction formula must be used 0 The reduction formulas are n sinn lx cosx n 1 n2 3m xdx 3m xdx n n n cosn lx Sinx n 1 n2 cos x dx cos x dx n n n tann lx n2 tan xdx tan xdx n 1 n secn zx tanx n 2 n2 sec x dx sec x dx n 1 n 1 0 Reduction formulas can be used on any trigonometric function of power greater than one to reduce it for integration Trigonometric Substitutions Trigonometric substitutions can be used for any function of the type a2X2 The X value can be substituted for either asin0 or atan0 Like u substitution the function must be scaled don t forget to complete dX equation for d0 The resulting function can then be solved in the same manner as a trigonometric equation The original X should be substituted back in after the integral is solved Partial Fractions Splitting up a more complicated integrand can make it easier to integrate When an integrand is in the form of two polynomials divided by one another they can be spilt using partial fractions The first step is to factor the polynomial in the denominator After that the entire equation is set equal to two or more fractions each with a single factor from the denominator The numerators are represented by ABC ect The three functions are then multiplied such that they have a common denominator which should be the same as the original function The entire numerator of the first function is then set equal to the numerator of the original equation The coefficient each power should then be set equal to the coefficient of the same power on the opposite side of the equation This should produce a system of equations The system can then be solved and the solution plugged into the fractions created at the beginning


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