https://r2.erweima.ai/mathbot/image/1726681993710ea6nuj95.jpg

The integral shown in the image is:

\[
\int e^x \cos x \, dx
\]

This type of integral requires **integration by parts** or can be solved using the method of **undetermined coefficients**. Here, I'll use the **integration by parts** technique.

We need to apply **integration by parts** twice, as the resulting expression will involve a similar integral. First, recall that:

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

### Step 1: First Integration by Parts
Let:
- \( u = \cos x \), so \( du = -\sin x \, dx \)
- \( dv = e^x \, dx \), so \( v = e^x \)

Using the integration by parts formula:

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

This simplifies to:

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

### Step 2: Second Integration by Parts
Now we need to integrate \( \int e^x \sin x \, dx \), applying integration by parts again. Let:
- \( u = \sin x \), so \( du = \cos x \, dx \)
- \( dv = e^x \, dx \), so \( v = e^x \)

Again applying the integration by parts formula:

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

### Step 3: Substituting Back
Now substitute this result into the first expression:

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

### Step 4: Solving for the Original Integral
Combine the integrals:

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

Move the integral on the right-hand side to the left-hand side:

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

Finally, divide by 2 to solve for the integral:

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

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

---

Would you like more details or have any questions about this solution?

Here are 5 related questions for further exploration:
1. How would you solve \( \int e^x \sin x \, dx \) on its own?
2. Can integration by parts be used to solve \( \int x e^x \, dx \)?
3. How do you determine when to use integration by parts?
4. What is the significance of the constant of integration?
5. How does the solution change if \( \cos x \) is replaced by \( \sin x \)?

**Tip**: When using integration by parts for oscillatory functions (like sine and cosine), always expect to use the method at least twice and combine similar terms to solve.

Learn how to find the integral of e^x cos(x) using integration by parts. This step-by-step solution covers key calculus concepts, including trigonometric and exponential functions, and demonstrates how to apply integration by parts twice to solve.

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