Let's consider the integrand given in the question:
[tex]\[
\int \sin(x) e^{\cos(x)} \, dx
\][/tex]
We need to find the integral of this expression. Here's a step-by-step explanation of how to approach this type of integral:
### Step-by-Step Solution
1. Identify the form of the integrand:
The integrand is [tex]\(\sin(x) e^{\cos(x)}\)[/tex]. Notice that this expression has both a trigonometric function ([tex]\(\sin(x)\)[/tex]) and an exponential function ([tex]\(e^{\cos(x)}\)[/tex]).
2. Use substitution:
A common approach for integrals involving trigonometric functions and exponentials is to use substitution. Let's set
[tex]\[
u = \cos(x)
\][/tex]
Then, taking the derivative of [tex]\(u\)[/tex] with respect to [tex]\(x\)[/tex],
[tex]\[
du = -\sin(x) \, dx \quad \Rightarrow \quad -du = \sin(x) \, dx
\][/tex]
3. Rewrite the integral:
Substitute [tex]\(u = \cos(x)\)[/tex] and [tex]\(du = -\sin(x) \, dx\)[/tex] into the integral.
[tex]\[
\int \sin(x) e^{\cos(x)} \, dx = \int e^u (-du)
\][/tex]
Simplifying,
[tex]\[
\int e^u (-du) = -\int e^u \, du
\][/tex]
4. Integrate:
The integral of [tex]\(e^u\)[/tex] with respect to [tex]\(u\)[/tex] is simply [tex]\(e^u\)[/tex].
[tex]\[
-\int e^u \, du = -e^u + C
\][/tex]
5. Substitute back to the original variable:
Recall that [tex]\(u = \cos(x)\)[/tex], so
[tex]\[
-e^u + C = -e^{\cos(x)} + C
\][/tex]
So, the integral of [tex]\(\sin(x) e^{\cos(x)} \, dx\)[/tex] is:
[tex]\[
- e^{\cos(x)} + C
\][/tex]
Given in the options is [tex]\(- e^{\cos(x)} e^{\sin(x)}\)[/tex], which can be interpreted like:
On comparing with the options provided, we see that:
- Option 2: [tex]\(-e^{\cos(x)}\)[/tex] appears as the correct integral.
- The function [tex]\(e^{\sin(x)} - e^{\cos(x)}\)[/tex].
Thus, the detailed answer is:
The integral of [tex]\(\sin(x) e^{\cos(x)} \, dx\)[/tex] is:
Option 2: [tex]\(-e^{\cos(x)}\)[/tex]
