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lec06 - (d Radicals(i √ ± x 2 ± a 2 use trigonometric...

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Z x n dx = x n +1 n + 1 ( n 6 = - 1) Z 1 x dx = ln | x | Z e x dx = e x Z a x dx = a x ln a 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 dx = ln | sin x | Z sinh x dx = cosh x Z cosh x dx = sinh x Z 1 x 2 + a 2 dx = 1 a tan - 1 x a Z 1 a 2 - x 2 dx = sin - 1 x a , a > 0 Z dx x 2 - a 2 = 1 2 a ln x - a x + a Z dx x 2 ± a 2 = ln x + p x 2 ± a 2
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Strategy for Integration 1. Simplify the Integrand if Possible - Try algebraic manipulations or trigonometric identities. 2. Look for an Obvious Substitution - Try to find a u = g ( x ) such that du = g 0 ( x ) dx also occurs. 3. Classify the Integrand According to Its Form (a) Trigonometric functions , use substitutions or half-angle formulas (b) Rational functions , use division and partial fractions (c) Integration by parts , look for products according to the LIATE rule
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Unformatted text preview: (d) Radicals (i) √ ± x 2 ± a 2- use trigonometric substitution (ii) n √ ax + b- use rationalization substitution u = n √ ax + b Strategy for Integration 4. Try Again (a) Try substitution , including non-obvious ones (b) Try parts , e.g. on single functions such as ln x (c) Manipulate the integrand , like in step 1. but more substantial (d) Relate the problem to previous problems , remember previously calculated integrals and previously used tricks (e) Use several methods , such as several substitutions, or substitutions and integration by parts...
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