The integral in the image is:

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

We are given $u = \sin(x)$, which suggests using a substitution approach to simplify the integral.

Solution:

1. Substitute $u = \sin(x)$:

   • Then, $\frac{du}{dx} = \cos(x)$, or $du = \cos(x) \, dx$.

2. Rewrite the integral in terms of $u$:

   • The integral now becomes: $\int e^{\sin(x)} \cos(x) \, dx = \int e^u \, du$

3. Integrate with respect to $u$:

   • The integral of $e^u$ with respect to $u$ is $e^u$.
   • So, we get: $e^u + C$

4. Substitute back $u = \sin(x)$:

   • The final answer is: $e^{\sin(x)} + C$

Final Answer:

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

Would you like more details, or do you have any questions?

Here are 5 related questions to deepen understanding:

1. How would you approach this integral if $e^{\sin(x)}$ was replaced with $e^{x^2}$?
2. What is the integral of $\sin(x) e^{\cos(x)} \, dx$ using a similar substitution?
3. How does substitution help simplify integrals involving trigonometric functions and exponentials?
4. Can you apply integration by parts to solve this integral? Why or why not?
5. How would you solve the integral $\int e^{\sin(x)} \sin(x) \, dx$?

Tip: For integrals involving exponential functions of trigonometric functions, look for a substitution that will simplify the exponent and the accompanying terms.