THE INTEGRATION OF sin n cos" 1 181 



To integrate sin 2 cos 4 



Now (2 sin 0) 2 (2 cos 0) 4 = (x - 1Y (x + -Y 



\ x/ \ x/ 



- 64 sin 2 cos 4 = (x 2 - i ) (x + -Y 

 V P/ \ / 



sin 2 cos 4 = - -{2cos60 + 4cos40-2cos20-4} 



Therefore sin 2 cos 4 dQ 

 J 



= ^r( 4 + 2 cos 20-4 cos 40-2 cos 60) d0 

 o4J 



= -J-J40 + sin 20 - sin 40 - i sin 60) 

 641 3 J 



= ^{l20 + 3 sin 20 - 3 sin 40 - sin 60 j 



EXAMPLES XII 

 Solve the following integrals : 



(1) f sec 2 x dx (2) Jcosec 2 x dx 



fsin 2 x dx fcos 2 x dx 



' ' J cos 4 x * ' J sin 4 x 



(5) [tan (ax + b) dx (6) (cot (ax + b) dx 



(7) I tan 5 a? da; (8) I tan 6 a; da; 



(9) [cot 3 x dx (10) [cot 4 x dx 



T dx j" dx_ 



' ' J 3 sin x + 4 cos x * ' J4 sin x 3 cos a; 



(18) f. ^ (14) f. * 



J sin x + cos a? J sin x cos x 



\3 sin 2 x+ 4 cos 2 a; J3sin 2 a;- 



4 cos 2 a; 



