Evaluating $\int e^{\cos(x)}\sin(x)\,dx$

Question

I trying to solve, but I find different solution of book. Someone can made a example step-by-step?

$$\int e^{\cos(x)}\sin(x)\,dx$$

Answer 1

Take $u = \cos x$ then $du/dx= -\sin x$

So $du =-\sin x dx $

Use the substitution, it becomes $$\int -e^u du = -e^u +c = -e^{\cos x} +c$$

Answer 2

Hint: Let $u=\cos(x)$, so that $du=-\sin(x)dx$.

Answer 3

Hint

Since $\frac{du}{dx}dx=du$, setting $\frac{du}{dx} =\sin x$ makes the most sense of we are substituting to solve. What is $\cos x$ in terms of $u$ when this is the case?