\displaystyle \int {2\,e^{\sin \left( x \right)}\,\cos \left( x \right)} \; \textrm{d}{x} =

To solve the integral \(\int 2e^{\sin(x)}\cos(x) \, \mathrm{d}x\), we can use the substitution method. Here's the step-by-step process:

1. **Substitution**: Let \( u = \sin(x) \). Then, differentiate \( u \) with respect to \( x \):
   \[
   \frac{du}{dx} = \cos(x) \implies du = \cos(x) \, dx
   \]

2. **Substitute**: Replace \(\sin(x)\) with \(u\) and \(\cos(x) \, dx\) with \(du\):
   \[
   \int 2e^{\sin(x)}\cos(x) \, \mathrm{d}x = \int 2e^u \, du
   \]

3. **Integrate**: The integral of \(2e^u\) with respect to \(u\) is:
   \[
   \int 2e^u \, du = 2e^u + C
   \]
   where \(C\) is the constant of integration.

4. **Back-substitute**: Replace \(u\) with \(\sin(x)\) to get the final answer:
   \[
   2e^{\sin(x)} + C
   \]

So the final result is:
\[
\int 2e^{\sin(x)}\cos(x) \, \mathrm{d}x = 2e^{\sin(x)} + C
\]

Would you like any further details or explanations? Here are some related questions:

1. How do you integrate \( e^{\sin(x)} \cos(x) \) without the coefficient of 2?
2. What is the integral of \( e^{\cos(x)} \sin(x) \)?
3. Can you explain the process of substitution more thoroughly?
4. How do you differentiate \( e^{\sin(x)} \)?
5. What if the exponent in the integral was \(\cos(x)\) instead of \(\sin(x)\)?

**Tip:** Substitution is particularly useful when the integral contains a function and its derivative. Identifying the right substitution can simplify the integral significantly.

Explore the integral of 2e^(sin(x)) cos(x) with a detailed step-by-step solution using the substitution method. Understand how to integrate complex trigonometric functions and exponential expressions. Suitable for undergraduate-level mathematics.

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