Question

Solve each of the integrals in Exercises 21–70. Some integrals require substitution, and some do not. (Exercise 69 involves a hyperbolic function.)

$\int \sin x{e}^{\cos x}\mathrm{d}x$

Short answer

The solution of the given integral is $\int \sin x{e}^{\cos x}\mathrm{d}x=-{\mathrm{e}}^{\mathrm{cosx}}$.

Step-by-step solution

Step 1. Given Information 

Solving the given integrals.

$\int \sin x{e}^{\cos x}\mathrm{d}x$

Step 2. Solving the given integral using substitution method. 

Let

$$\begin{gathered} u=\cos x \\ \frac{du}{dx}=-\sin x \\ -du=\sin xdx \end{gathered}$$

Step 3. This substitution changes the integral into  

$$\begin{gathered} \int \sin x{e}^{\cos x}\mathrm{d}x=-\int e^u\mathrm{du} \\ \int {\mathrm{sinxe}}^{\mathrm{cosx}}\mathrm{dx}=-{\mathrm{e}}^{\mathrm{u}} \\ \int {\mathrm{sinxe}}^{\mathrm{cosx}}\mathrm{dx}=-{\mathrm{e}}^{\mathrm{cosx}} \end{gathered}$$