https://file.aiquickdraw.com/mathbot/file/17278593269779emcirh7.jpg

The integral provided in the image is:

\[
\int e^x \sin(x) \, dx
\]

To solve this integral, we can use the method of **integration by parts** twice. The formula for integration by parts is:

\[
\int u \, dv = uv - \int v \, du
\]

Let’s proceed step by step:

1. First, we set:
   - \( u = \sin(x) \), which gives \( du = \cos(x) \, dx \)
   - \( dv = e^x \, dx \), which gives \( v = e^x \)

Now, apply the integration by parts formula:

\[
\int e^x \sin(x) \, dx = e^x \sin(x) - \int e^x \cos(x) \, dx
\]

2. For the second integral \( \int e^x \cos(x) \, dx \), we apply integration by parts again. This time, we set:
   - \( u = \cos(x) \), which gives \( du = -\sin(x) \, dx \)
   - \( dv = e^x \, dx \), which gives \( v = e^x \)

Thus, applying integration by parts again:

\[
\int e^x \cos(x) \, dx = e^x \cos(x) - \int e^x (-\sin(x)) \, dx
\]

Simplifying the expression:

\[
\int e^x \cos(x) \, dx = e^x \cos(x) + \int e^x \sin(x) \, dx
\]

3. Now we substitute this result back into the first equation:

\[
\int e^x \sin(x) \, dx = e^x \sin(x) - \left( e^x \cos(x) + \int e^x \sin(x) \, dx \right)
\]

4. Combine like terms:

\[
\int e^x \sin(x) \, dx - \int e^x \sin(x) \, dx = e^x \sin(x) - e^x \cos(x)
\]

This simplifies to:

\[
0 = e^x (\sin(x) - \cos(x)) + \int e^x \sin(x) \, dx
\]

5. Solve for \( \int e^x \sin(x) \, dx \):

\[
2 \int e^x \sin(x) \, dx = e^x (\sin(x) - \cos(x))
\]

Thus:

\[
\int e^x \sin(x) \, dx = \frac{e^x (\sin(x) - \cos(x))}{2} + C
\]

where \( C \) is the constant of integration.

### Final Answer:

\[
\int e^x \sin(x) \, dx = \frac{e^x (\sin(x) - \cos(x))}{2} + C
\]

Would you like more details on any step or have any further questions?

### Related Questions:
1. How does integration by parts work in different cases?
2. Can integration by parts be applied to trigonometric integrals in other forms?
3. What is the role of constants of integration in indefinite integrals?
4. How do we identify when to use integration by parts versus other methods of integration?
5. Can we solve similar integrals involving \(e^x\) and different trigonometric functions?

### Tip:
Always verify the conditions under which specific integration techniques, like parts or substitution, simplify the problem for more complex functions.

Solve the integral ∫e^x sin(x) dx using the method of integration by parts. Detailed step-by-step solution involving calculus concepts and multiple applications of the integration by parts formula.

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