integrate 10*e^5t*cos5t.
Do a partial integration twice, e.g. integrate the exponential function, and then in the integral of e^(5 t) sin(5 t), you again integrate the e^( 5 t) factor (and not the sin(5t) ). You then get the the same integral back but with additional terms. You then solve for the desired integral by bringing it to one side of the equation.
Alternatively and much faster, if you are familiar with complex numbers, you can write the integral as the real part of the integral of exp[5(1+i)t], which you can integrate directly.