∫_-a^ae^axcosnxdx

see , you have to  use by parts.. 

.. for simplification put “ nx “ = “t” … then dx = dt/n , and e^ax = e^(a/n)*t

 ^{now ,  Take  COSt as the first part and} e^{(a/n)*t as the second part ..} 

 I =∫_-a^ae^axcosnxdx

 

I =( cost ) * ( e^(a/n)*t  )*( n/a )  +  ∫( sint )  * (  e^(a/n)*t )* ( n/a )dt   ------------> {1}

                                         = again taking ( sint) as the first part and the other one as second part  we can write the later term including integration sign as  

  ( sint ) *  (  e^(a/n)*t )* ( n/a )* ( n/a ) –  ∫(cost) *  (  e^(a/n)*t )* ( n/a )* ( n/a ) dt

 = ( sint ) *  (  e^(a/n)*t )* ( n/a )* ( n/a )  – I* ( n/a )* ( n/a )  

  now put this  value in  equation number {1} . and then solve urselve to get the answer  

(a² /n)*(1/ (a² + n² ) )*[e^ax ] * ( acosnx + nsinnx )