Math191_Week11_section - a. R xdx 4+ x 2 (1) 1 2 R d (4+ x...

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Math 191 Problem Set for Week 10 Section 1. Evaluate the following integrals using integration by parts: a. R cos - 1 ( x 2 ) dx b. R e - 2 x sin 3 xdx 2. Evaluate the integrals. It may be necessary to use a substitution first. a. R dx x (1+ 3 x ) u = 3 x du = dx 3 x 2 / 3 dx = 3 u 2 du R 3 u 2 du u 3 (1+ u ) = 3 R du u (1+ u ) = 3 ln ± ± ± u u +1 ± ± ± + C = 3 ln ± ± ± 3 x 1+ 3 x ± ± ± + C b. R ds e s +1 u = e s + 1 du = e s ds 2 e s +1 ds = 2 udu u 2 - 1 R 2 udu u ( u 2 - 1) = 2 R du ( u +1)( u - 1) = R du u - 1 - R du u +1 = ln ± ± ± u - 1 u +1 ± ± ± + C = ln ± ± ± e s +1 - 1 e s +1+1 ± ± ± + C 3. Evaluate the integrals (1) without using a trigonometric substitution, (2) using a trigonometric substi- tution.
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Unformatted text preview: a. R xdx 4+ x 2 (1) 1 2 R d (4+ x 2 ) 4+ x 2 = 4 + x 2 + C (2) [ x = 2 tan y ] R 2 tan y 2 sec 2 ydy 2 sec y = 2 R sec y tan ydy = 2 sec y + C = 4 + x 2 + C b. R tdt 4 t 2-1 (1) 1 8 R d (4 t 2-1) 4 t 2-1 = 1 4 4 t 2-1 + C (2) [ t = 1 2 sec ] R 1 2 sec tan 1 2 sec d tan = 1 4 R sec 2 d = tan 4 + C = 4 t 2-1 4 + C 4. R e t tan 2 e t + 1 dt 1...
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