Hello !
I propose another solution, shorter (?), using complex numbers.
#e^{6x} cos(5x) = e^(6x)\mathfrak{Re}(e^{5ix}) = \mathfrak{Re}(e^{(6+5i)x})#.
Now integrate easily :
#int e^{6x} cos(5x)dx = \mathfrak{Re} \int e^{(6+5i)x} d x = \mathfrak{Re} (e^((6+5i)x))/(6+5i) + c#
where #c \in CC# is a constant.
To take the real part, you have to write :
#(e^{(6+5i)x})/(6+5i) = e^(6x)((cos(5x) + i sin(5x))(6-5i))/((6+5i)(6-5i)) = e^(6x)((cos(5x) + i sin(5x))(6-5i))/(36+25)#
so,
#\mathfrak{Re} (e^((6+5i)x))/(6+5i) = e^(6x)(6 cos(5x) + 5 sin(5x))/61#.
Finally,
#int e^{6x} cos(5x) dx =e^(6x)(6/61 cos(5x) + 5/61 sin(5x))+c#
