Integrate by parts:
#int 5^x sinx dx = int 5^x d/dx (-cosx) dx #
#int 5^x sinx dx = = -5^xcosx + int cosx d/dx(5^x) dx #
#int 5^x sinx dx = = -5^xcosx + ln5 int cosx 5^xdx #
and then again:
#int 5^x sinx dx = = -5^xcosx + ln5 int d/dx(sinx) 5^xdx #
#int 5^x sinx dx = = -5^xcosx + ln5sinx5^x - ln5 int sinx d/dx( 5^x)dx #
#int 5^x sinx dx = = 5^x(ln5sinx- cosx) - ln^2 5 int sinx5^xdx #
The integral now appears on both sides of the equation and we can solve for it:
#(1+ln^2 5)int 5^x sinx dx = 5^x(ln5sinx- cosx)+C#
#int 5^x sinx dx = (5^x(ln5sinx- cosx))/(1+ln^2 5)+C#
