190 PRACTICAL MATHEMATICS 



Then le 4 * sin 2 x cos 3 x dx 



= -I e^ cos x dx I e** cos 3x dx I e* x cos 5x dec 

 8J 16J loj 



g4x g4z gix 



= 7= cos (* a) cos (Sx B) -- 7= cos (5# v) 

 8A/17 80 16\/41 



13 5 



where tan a = 7, tan 3 = -, and tan y = - 

 44 4 



p e 4 * r 2 / i\ 



Hence I e* 31 sin 2 x cos 3 x dx = \ /" cos ( tan -1 -) 

 J 16lVl7 V 4 / 



- cos ( 3x tan -1 - ) -- 7= cos ( 5x tan" 1 - 

 5 \ 4/ V41 \ 4 



EXAMPLES XIII 

 Solve the following integrals : 



(1) \x 7 log a # dx (2) I \/x logga; dx 



(5) fa? (log^r) 2 <to (6) JV* 



(7) f^e^da; (8) (x 2 e~^ dx 



(9) f^ ina: cos 3 x dx (10) fe staa! sin 2x dx 



(put sin x = 2) (put sin a; = 2) 



(11) f 2 sin 2x dx (12) frc 3 cos 3x dx 



(13) U 4 sin 2# dx (14) j 2 sin 2 x dx 



(15) U 3 cos 2 x dx (16) fa; 2 sin 3 x 



(17) fcT 2 cos 3 x dx (18) 



(19) U 2a! cos 3a; dx (20) U" 331 sin 2x dx 



dx 

 e 2 * sin 3a; dx 



