M151_Sp02_sols - MATH 151 CALCULUS II Group Final(Spring...

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MATH 151 — Group Final ( Spring 2002 ) SOLUTIONS — page 1 of 7 1 Compute the indefinite integral ( ) cos x x dx 5 # . We will determine this integral using Integration by Parts, uv v du u dv = - # # . The table below shows how we choose u and dv for the given integral, along with the corresponding du and v : u = x v = ( ) sin x 5 5 1 du = dx dv = ( ) cos x 5 dx Applying the Integration by Parts formula, uv v du u dv = - # # , we find that ( ) ( ) ( ) ( ) ( ) sin sin cos sin cos dx x x dx x x x C x x x 5 5 5 5 5 5 1 5 1 5 1 25 1 = - = + + # # 2 Compute the indefinite integral ( ) ( ) cos sin dx x x 2 3 # . First, we rewrite ( ) sin x 3 as ( ) ( ) sin sin x x 2 , ( ) ( ) ( ) ( ) ( ) cos sin cos sin sin x x dx x x x dx 2 3 2 2 = # # and then we use the trig identity ( ) ( ) sin cos x x 1 2 2 / - : ( ) ( ) ( ) ( ) ( ) ( ) cos cos sin cos cos sin x x x dx x x x dx 1 2 2 2 4 = - = - 7 7 A A # # We now make the substitution u = cos ( x ), du = -sin ( x ) dx : ( ) ( ) ( ) ( ) ( ) ( ) ( ) cos sin cos cos u u du u u du u u x x dx x x C C 2 4 4 2 5 1 5 3 1 3 2 3 5 1 5 3 1 3 ` = - - = - = - + = - + # # # M ATH 151: C ALCULUS II Group Final ( Spring 2002 ) SOLUTIONS by Steve Leduc
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MATH 151 — Group Final ( Spring 2002 ) SOLUTIONS — page 2 of 7 3 Compute the indefinite integral ( )( ) x x dx x 3 1 19 + - - # This is a rational function which we will integrate by first performing a partial - fraction decomposition: ( )( ) x x x x A x B 3 1 19 3 1 - = + + - + - To find the constants A and B that make this equation an identity, we begin by multiplying both sides by ( x - 3)( x + 1), to clear all the fractions: ( )( ) ( )( ) ( )( ) ( ) ( ) x x x x x x x x A x B x A x B x (*) 3 1 3 1 19 3 1 3 1 19 1 3 - + - + - = - + - + + - = + + - ; E We may now use the following technique: By setting x = 3 in Equation (*), we get ( ) ( ) ( ) A
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This note was uploaded on 05/24/2008 for the course MATH 151 taught by Professor Geveci during the Spring '08 term at San Diego State.

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M151_Sp02_sols - MATH 151 CALCULUS II Group Final(Spring...

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