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34. Volume 5 E 1 0 2 p x 3 } ( x 1 1) 2 (2 2 x ) } 4 dx 5 4 p E 1 0 } ( x 1 1 x ) d (2 x 2 x ) } } ( x 1 1) x (2 2 x ) } 5 } x 1 A 1 } 1 } 2 2 B x } x 5 A (2 2 x ) 1 B ( x 1 1) 5 ( 2 A 1 B ) x 1 (2 A 1 B ) Equating coefficients of like terms gives 2 A 1 B 5 1 and 2 A 1 B 5 0 Solving the system simultaneously yields A 5 2 } 1 3 } , B 5 } 2 3 } . 4 p E 1 0 } ( x 1 1 x ) d (2 x 2 x ) } 5 } 4 3 p } 1 E 1 0 } x 2 1 dx 1 } 1 E 1 0 } 2 2 2 dx x } 2 5 } 4 3 p } 13 2 ln ) x 1 1 ) 4 1 3 2 2 ln ) 2 2 x ) 4 2 5 } 4 3 p } ( 2 ln 2 1 2 ln 2) 5 } 4 p 3 ln 2 } 35. y 5 3 tan u , dy 5 3 sec 2 u d u , 2 } p 2 } , u , } p 2 } 9 1 y 2 5 9 1 9 tan 2 u 5 9 sec 2 u E } ˇ 9 w d 1 w y y w 2 w } 5 E } 3 ) 3 s s e e c c 2 u u ) } d u 5 E sec u d u 5 ln ) sec u 1 tan u ) 1 C 1 5 ln ) } ˇ 9 w 3 1 w y w 2 w } 1 } 3 y } ) 1 C 1 5 ln ) ˇ 9 w 1 w y w 2 w 1 y ) 2 ln 3 1 C 1 5 ln ) ˇ 9 w 1 w y w 2 w 1 y ) 1 C Use Formula 88 for E sec u d u with x 5 u and a 5 1. Use Figure 8.18(a) from the text with a 5 3 to get ˇ 9 w 1 w y w 2 w 5 3 ) sec u ) . 36. t 5 5 sin u , dt 5 5 cos u d u , 2 } p 2 } # u # } p 2 } 25 2 t 2 5 25 2 25 sin 2 u 5 25 cos 2 u E ˇ 2 w 5 w 2 w t w 2 w dt 5 E 5 ) cos u ) (5 cos u ) d u 5 25 E cos 2 u d u 5 25 E } 1 1 c 2 os 2 u } d u 5 } 25 2 u } 1 } 25 s 4 in 2 u } 1 C 5 } 2 2 5 } u 1 } 2 2 5 } sin u cos u 1 C 5 } 2 2 5 } sin 2 1 } 5 t } 1 } 2 2 5 } 1 } 5 t } 2 } ˇ 2 w 5 w 5 2 w t w 2 w } 1 C 5 } 2 2 5 } sin 2 1 } 5 t } 1 } t ˇ 2 w 5 w 2 2 w t w 2 w } 1 C Use Figure 8.18(b) from the text with a 5 5 and x 5 t to get ˇ 2 w 5 w 2 w t w 2 w 5 5 ) cos u ) . 37. x 5 } 7 2 } sec u , dx 5 } 7 2 } sec u tan u d u , 0 # u , } p 2 } 4 x 2 2 49 5 49 sec 2 u 2 49 5 49 tan 2 u E } ˇ 4 w x d 2 w x 2 w 4 w 9 w } 5 E } 7/2 se ) 7 c t u an ta u n ) u d u } 5 E } 1 2 } sec u d u 5 } 1 2 } ln ) sec u 1 tan u ) 1 C 1 5 } 1 2 } ln ) } 2 7 x } 1 } ˇ 4 w x 2 w 7 2 w 4 w 9 w } ) 1 C 1 5 } 1 2 } ln ) 2 x 1 ˇ 4 w x 2 w 2 w 4 w 9 w ) 2 } 1 2 } ln 7 1 C 1 5 } 1 2 } ln ) 2 x 1 ˇ 4 w x 2 w 2 w 4 w 9 w ) 1 C Use Figure 8.18(c) from the text with a 5 } 7 2 } to get ! x 2 § 2 § } 4 § 4 9 } § 5 } 7 2 } tan u . 1 0 1 0 Section 8.4 351

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38. x 5 tan u , dx 5 sec 2 u d u , 2 } p 2 } , u , } p 2 } x 2 1 1 5 tan 2 u 1 1 5 sec 2 u E } x 2 ˇ d x x 2 w 1 w 1 w } 5 E } ta s n e 2 c u 2 u ) se d c u u ) } 5 E } se ta c n u 2 d u u } 5 E } s c i o n s 2 u u } d u 5 E (sin u ) 2 2 cos u d u 5 2 (sin u ) 2 1 1 C 5 2 csc u 1 C 5 2 } ˇ 1 w x 1 w x w 2 w } 1 C Use Figure 8.18(a) from the text with a 5 1 to get csc u 5 } ˇ 1 w x 1 w x w 2 w } .
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