INTEGRATION BY PARTS 187 



/ ,r r> fM- 3 \ .->/'. 15 (I , f , 



( --- 1 -- )cos 2.1'H -- sin 2a; -{-a; 2 sin 2x \x sm 2x dx 

 \ - - / 4 2 vs J 



/5a^ 15aA 



2 2 VT~ TV s 



i 



a- 5 5r a \ /5a^ 15aA 15 



sm 2x+ ~ ^ cos 



4 



107. (4) To integrate ff** sin bx and e?* cos bx. 

 Now le * sin bx dx uv \vdu 



where u = sin bx, du= b cos fa da; 



r c"* 



and di> = eP* dx, v = le"* da; = 



J a 



f 1 6 f 



Then I e 035 sin fa da; = - e * sin fa I e * cos fa da;, and 



J J 



denoting I e?* sin fa dx by X and I &* cos fa da; by Y, 



X = - e?* sin fa Y . (1) 



a a 



Also 1^ cos bx dx = uv \v du 



where u = cos fa, dw = b sin fa da; 



r ^Q* 



and du = e?* dx, v = \e/ ax dx = 



J a 



f 1 b f 



Then le"* cos bxdx = - a * cos fa + - le * sin fa da; 

 J a aj 



or Y=-c aa! cosfa+-X (2) 



a a 



Thus giving a pair of simultaneous equations to be solved for 

 X and Y. 



1 t 70 



Solving for X, X = - e * sin fa % ft cos fa 3 X 



a a 2 a 2 



X (a 2 + fe 2 ) = ^{a sin fo - b cos fa} 



gOZ 



X = z p (a sin bx b cos bx } 



P gate j 



or \e/ ax sin bx dx = . sin (for a) where tan a = - 



J Va* + b 2 a 



