Reconciling specific differentiation results with anti-derivatives

Hi,

I'm stuck with the following question:

Differentiate e^(-t/50)*sin(π*t/5) and  e^(-t/50)*cos(π*t/5). Use the results to find an antiderivative of e^(-t/50)*sin(π*t/5).

I've found the two derivatives, but am stuck now. Any help would be appreciated.

Thanks!

Student X

d/dt [e^(-t/50)*sin(π*t/5)] = -1/50*e^(-t/50)*sin(π*t/5)+π/5*e^(-t/50)*cos(π*t/5)    --------------(1) 

d/dt [e^(-t/50)*cos(π*t/5)] = -1/50*e^(-t/50)*cos(π*t/5)- π/5*e^(-t/50)*sin(π*t/5)   --------------(2)

Multiplying (1) by 1/(10π) on both sides, 

we have  

1/(10π)*d/dt [e^(-t/50)*sin(π*t/5)]

= -1/(500π) *e^(-t/50)*sin(π*t/5)+1/50*e^(-t/50)*cos(π*t/5) ---------------------------------(3)

Add (2) to (3), we have       

d/dt [ e^(-t/50)*cos(π*t/5)+1/(10π)*e^(-t/50)*sin(π*t/5)] 

= [-1/(500π)- π/5 ]*e^(-t/50)*sin(π*t/5)

= (-1- 100π ^2) /(500π)*e^(-t/50)*sin(π*t/5)       

= -(1+ 100π ^2) /(500π)*e^(-t/50)*sin(π*t/5)       

Taking the anti-derivative on both sides (aka integrating both sides wrt t): 

e^(-t/50)*cos(π*t/5)+1/(10π)*e^(-t/50)*sin(π*t/5) 

= -(1+100π ^2) /(500π)* [anti-derivative of e^(-t/50)*sin(π*t/5) ] 

Hence, the anti-derivative of e^(-t/50)*sin(π*t/5)  

= -[(500π)/(1+ 100π ^2)]*[e^(-t/50)*cos(π*t/5)+ 1/(10π) *e^(-t/50)*sin(π*t/5)]

Hope this helps. Peace.

Best Regards,

Mr Koh