Spring 2014 - Formulas for Test 1 - Summary for test1 1...

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Summary for test1 1. Integration Formula (constants are omitted) Z x n dx = x n +1 n + 1 ( n 6 = - 1) Z e x dx = e x Z 1 x dx = ln | x | Z sin x dx = - cos x Z cos x dx = sin x Z sec 2 x dx = tan x Z csc 2 x dx = - cot x Z sec x tan x dx = sec x Z csc x cot x dx = - csc x Z sec x dx = ln | sec x + tan x | Z csc x dx = ln | csc x - cot x | Z tan x dx = ln | sec x | Z cot x = ln | sin x | Z 1 x 2 + a 2 = 1 a tan - 1 x a 2. Trigonometric Identities sin 2 x + cos 2 x = 1 sec 2 x = 1 + tan 2 x sin 2 x = 1 2 (1 - cos 2 x ) cos 2 x = 1 2 (1 + cos 2 x ) 3. Trigonometric Substitutions a 2 - x 2 : substitute x = a sin θ x 2 + a 2 : substitute x = a tan θ x 2 - a 2 : substitute x = a sec θ 4. Partial Fractions Consider a rational function P ( x ) Q ( x ) . i) If Q ( x ) contains ( ax + b ) n , then the partial fractions decomposition contains A 1 ax + b + A 2 ( ax + b ) 2 + · · · + A n ( ax + b ) n . ii) If Q ( x ) contains ( ax 2 + bx + c ) n (where ax 2 + bx + c is irreducible), then the partial fractions decomposition contains A 1 x + B 1 ax 2 + bx + c + A 2 x + B 2 ( ax 2 + bx + c ) 2 + · · · + A n x + B n ( ax 2 + bx + c ) n .

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