\[I = \int\limits_0^{\frac{\pi }{2}} {sinx.co{s^3}x({e^{si{n^2}x}} + 1)} dx\]Đặt $t = si{n^2}x \Rightarrow \frac{{dt}}{2} = \sin x\cos xdx$
$\Rightarrow I = \frac{1}{2}\int\limits_0^1 {(1 - t)({e^t} + 1)dt}$
Đặt $u = 1 - t \Rightarrow du = - dt$
$dv = ({e^t} + 1)dt \Leftarrow v = {e^t} + t$
$ \Rightarrow I = \frac{1}{2}(1 - t)({e^t} + t)\begin{cases}1 \\ 0 \end{cases} + \frac{1}{2}\int\limits_0^1 {({e^t} + t)dt} $
$ \Leftrightarrow I = - \frac{1}{2} + \frac{1}{2}({e^t} + \frac{{{t^2}}}{2})\begin{cases}1 \\ 0 \end{cases} = \frac{1}{4}(2e - 3)$