# How to make a substitution and then use integration by parts to evaluate the integral.∫ e^(cost) sin2t dt ?

$- 2 \left[\cos t {e}^{\cos t} - {e}^{\cos t}\right] + c$
$y = \cos t$
$\mathrm{dy} = - \sin t \mathrm{dt}$
${e}^{\cos} t \sin 2 t = {e}^{y} \left(- 2 y\right) \mathrm{dy}$
$\int {e}^{\cos} t \sin 2 t = \int {e}^{y} \left(- 2 y\right) \mathrm{dy} = - 2 \int {e}^{y} y \mathrm{dy} = - 2 \left[y {e}^{y} - {e}^{y}\right] + c = - 2 \left[\cos t {e}^{\cos t} - {e}^{\cos t}\right] + c$