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Unformatted text preview: M10360 Exam 3: Review Guide Chapter 8 8.3 Trigonometric Integrals Know the main identities: 1. sin 2 x + cos 2 x = 1 2. sin 2 x = 1- cos(2 x ) 2 3. cos 2 x = 1 + cos(2 x ) 2 4. tan 2 x + 1 = sec 2 x Understand how to split off cosines, sines, secants and tangents to be able to use one of the above identities along with a u-substitution. 8.5 Partial Fractions Know how to split up a fraction into its partial fraction decomposition. Follow the steps given below: 1. Make sure the numerator has smaller degree than the denominator. Otherwise, do long division. 2. Remember to factor the denominator completely . Combine linear and irre- ducible quadratic terms. Repeated linear terms are of the form ( px + q ) m . Repeated irreducible quadratic terms are of the form ( ax 2 + bx + c ) n . A quadratic is irreducible if b 2- 4 ac < 0. 3. For each repeated linear term ( px + q ) m , we get m fractions in the decomposition of the form: A ( px + q ) + B ( px + q ) 2 + + D ( px + q ) m (notice linear terms, even though repeated, get constants in the numerator) and for each repeated irreducible quadratic term ( ax 2 + bx + c ) n , we get n fractions in the decomposition of the form Ax + B ( ax 2 + bx + c ) + Cx + D ( ax 2 + c ) 2 + + Ex + F ( ax 2 + bx + c ) n (notice irreducible quadratic terms all get linear terms in the numerator). After doing the decomposition, re-add the fractions and set the numerators equal. This will give you a bunch of equations based on coefficients of x,x 2 and the constant term. Use these equations to solve for the unknown parameters, A , B , C , etc....
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