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Unformatted text preview: Last Time: Two Techniques of Integration Last time we talked about two tools for integrating difficult functions: I Integration by Substitution (Change of Variable) Z x = b x = a f ( u ( x )) dx = Z u = u ( b ) u = u ( a ) f ( u ) dx du du = Z u = u ( b ) u = u ( a ) f ( u ) x ( u ) du = Z u = u ( b ) u = u ( a ) f ( u ) u ( x ) du I Integration by Parts Z udv = uv Z vdu Z b a udv = uv b a Z b a vdu Remember the following “tips” we talked about: I If you see a compound function, think about trying a substitution. Eg., Z 2 1 e 3 x dx = Z 6 3 e u 1 3 du = 1 3 Z 6 3 e u du I If you’re integrating the product of a polynomial times a function that antidifferentiates nicely (like e x or sin( x ) or cos( x )), try integration by parts. Eg., Z xe x dx = xe x Z e x dx = xe x e x + C I If you have a polynomial times ln( x ), ignore the last tip— differentiate the ln( x ) and antidifferentiate the polynomial. Eg., Z x 2 ln( x ) = x 3 3 ln( x ) Z x 3 3 1 x dx = 1 3 x 3 ln( x ) 1 3 Z x 2 dx Polynomial Substitutions— R 1 x √ 4 x 2 dx First, notice the composite function in the denominator, √ 4 x 2 . This immediately suggests trying the substitution u = 4 x 2 , which gives du = 2 xdx . Fortunately, we already have the requisite 2 x in the numerator! (once we play with constants...) Z 1 x √ 4 x 2 dx = 1 2 Z 1 2 x √ 4 x 2 dx = 1 2 Z x =1 x =0 1 √ 4 x 2 · ( 2 xdx ) = 1 2 Z u =3 u =4 1 √ u du = 1 2 Z u =3 u =4 u 1 / 2 du = 1 2 2 u 1 / 2 u =3 u =4 = (3 1 / 2 4 1 / 2 ) = 2 √ 3 WARNING: As we already saw in the case of R 2 √ 4 x 2 dx , this “first instinct” substitution is not necessarily going to work! Z x =2 x =0 p 4 x 2 dx = Z u =0 u =4 √ udx but there’s no good way to change from “ dx ” to “ du ” without an extra x factor. *Remember: Adding (or removing) constants (like 2) is easy— but you can’t add or remove functions! Hidden Substitutions— R tan( x ) dx First, use a trigonometric identity to write Z tan( x ) dx = Z sin( x ) cos( x ) dx Now look at what we have: the expression looks more complicated, but the sin( x ) in the numerator is perfectly suited for the substitution u = cos( x ), since this means du = sin ( x ) dx . Z tan ( x ) dx = Z sin( x ) cos( x ) dx = Z 1 cos( x ) ( sin( x ) dx ) = Z 1 u du = ln  u  + C = ln  cos( x )  + C = ln  sec( x )  + C Sines and Cosines— R sin 3 ( x ) cos 2 ( x ) dx We know how to integrate functions like sin n ( x ) cos( x ), or sin( x ) cos m ( x ), by splitting off the solitary sin( x ) or cos( x ) to make a substitution. For example, Z sin 3 ( x ) cos( x ) dx = Z sin 3 ( x ) · (cos( x ) dx ) = Z u 3 du = 1 4 u 4 + C = 1 4 sin 4 ( x ) + C by making the substitution u = sin( x ) (so that du = cos( x ) dx )....
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This note was uploaded on 02/10/2010 for the course MAT 132 taught by Professor Poole during the Spring '08 term at SUNY Stony Brook.
 Spring '08
 POOLE
 Calculus, Integration By Substitution

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