$f(x)=e^{\cos^2x}.\cos^3x.\sin x\\ t=\cos^2x \Rightarrow dt=-2\sin x\cos x\\ I=\displaystyle\int f(x) \, dx\\ =\dfrac{1}{2}\displaystyle\int e^{\cos^2x}.\cos^2x.2\cos x.\sin x \, dx\\ =-\dfrac{1}{2}\displaystyle\int e^t.t \, dt\\ u=t \Rightarrow du=dt\\ dv=e^tdt \Rightarrow v=e^t\\ I=-\dfrac{1}{2}\left(te^t-\displaystyle\int e^t \,dt\right)\\ =-\dfrac{1}{2}\left(te^t-e^t \right)+C\\ \Rightarrow A$
