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# prfsol - Math 115 — Final Practice Answers 1(a Substitute...

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Unformatted text preview: Math 115 — Final Practice Answers 1. (a) Substitute u = sin( x ), obtaining e sin( x ) + C . (b) Integrate by parts twice, obtaining- x 2 cos( x ) + 2 x sin( x ) + 2 cos( x ) + C . (c) Substitute u = ln x , obtaining 23 on the table of integrals. The result is- radicalbig 1 + (ln x ) 2 ln x + C . 2. (a) Note 0 ≤ cos 2 ( x ) 1 + x 2 ≤ 1 1 + x 2 . The antiderivative of 1 1 + x 2 is arctan( x ), so the integral converges to π 2- (- π 2 ) = π and the original integral converges. (b) For x > 0 recall sin( x ) < x and so 1 / sin( x ) > 1 /x . Because the antiderivative of 1 /x is ln x , the integral integraltext π/ 2 dx/x diverges. Consequently, integraltext π/ 2 dx/ sin( x ) diverges by comparison. 3. Use the method of shells. Then Volume = integraltext 1 2 πxe x 2 dx = πe x 2 | 1 = π ( e- 1). 4. (a)-1-0.5 0.5 1 1.5 2-1-0.5 0.5 1 (b) The curves intersect where 1 = 1 + cos( θ ), hence θ = π 2 , 3 π 2 . Then Area = integraldisplay 3 π/ 2 π/ 2 1 2 (1 2- (1 + cos( θ )) 2 dθ . Using cos 2 ( θ ) = 1 2 (1 + cos(2 θ )), the integral evaluates to 2- π 4 ....
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prfsol - Math 115 — Final Practice Answers 1(a Substitute...

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