quiz4ws2009 - x and cos x into an ordinary rational...

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Quiz 4 Instructor: Stephan Spencer 24 February, 2009 Student Name: 1. Use trig substitution to compute the following integrals a). Z x 3 x 2 + 100 dx b). Z a 0 x 2 p a 2 - x 2 dx c). Z 1 x 5 9 x 2 - 1 dx d). Z 1 t 2 - 6 t + 13 dt e). Z π/ 2 0 cos t p 1 + sin 2 t dt f). Z x 0 p a 2 - t 2 dt 2. Use partial fractions to compute the following integrals. a). Z 1 0 x 3 - 4 x - 10 x 2 - x - 6 dx b). Z 1 s 2 ( s - 1) 2 ds c). Z 3 x 2 + x + 4 x 4 + 3 x 2 + 2 dx d). Z 1 0 x x 2 + 4 x + 13 dx 1
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3. Compute the following integrals. a). Z x csc x cot x dx b). Z 4 0 x - 1 x 2 - 4 x - 5 dx c). Z ln x x p 1 + (ln x ) 2 dx d). Z 1 0 (1 + x ) 8 dx e). Z 1 x + 3 x dx f). Z sec θ tan θ sec 2 θ - sec θ 4. The German mathematician Karl Weierstrass (1815-1897) noticed that the substitution t = tan( x/ 2) will convert any rational function of sin
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Unformatted text preview: x and cos x into an ordinary rational function of t . a). If t = tan( x/ 2) ,-π < x < π , show that cos( x/ 2) = 1 √ 1+ t 2 and sin( x/ 2) = t √ 1+ t 2 . b). Use part a) to show that cos( x ) = 1-t 2 1+ t 2 and sin( x ) = 2 t 1+ t 2 . c). Show that dx = 2 1+ t 2 dt . 5. Use the method outlined in the previous problem to integrate the follow-ing: a). Z 1 3-5 sin x dx b). Z π/ 2 π/ 3 1 1 + sin x-cos x dx 2...
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This note was uploaded on 11/29/2010 for the course MATH 1700 taught by Professor Staff during the Spring '08 term at Missouri (Mizzou).

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quiz4ws2009 - x and cos x into an ordinary rational...

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