3.6 Formula Sheet - Formula sheet for Prelim 3 in Math 106 on Tuesday Antiderivatives(indefinite integrals xp dx = xp 1 p 1 for p =-1 x-1 dx = ln

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Formula sheet for Prelim 3 in Math 106 on Tuesday April 24, 2007 Antiderivatives (inde±nite integrals) ± x p dx = x p +1 p +1 for p ± = - 1 ± x - 1 dx =ln | x | ± e rx dx = e r Substitution. Suppose G ± = g . Letting u = f ( x ), du = f ± ( x ) dx ± g ( f ( x )) f ± ( x ) dx = ± g ( u ) du = G ( u )= G ( f ( x )) Integration by parts ± f ( x ) g ± ( x ) dx = f ( x ) g ( x ) - ± f ± ( x ) g ( x ) dx Defnite integrals. The area under the curve f between a and b is given by ± b a f ( x ) dx = F ( b ) - F ( a F ( x ) | b a where F is ANY antiderivative. If we rotate the part of the curve f over the interval [ a, b ] around the x -axis, the result is a solid oF revolution with volume V = ² b a πf ( x ) 2 dx . The improper integral ± 0 f ( x ) dx = lim b →∞ ± f ( x ) dx To ±nd the limit it is useful to know that if p, q, r > 0 then as x →∞ e - ,x - q q e - , (ln x ) - p , (ln x ) p x - q 0 A probability density has f ( x ) 0 and ² f ( x ) dx = 1. The mean μ = EX = ² xf ( x ) dx . The second moment
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This note was uploaded on 11/14/2007 for the course MATH 1106 taught by Professor Durrett during the Spring '07 term at Cornell University (Engineering School).

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