To solve the integral \( \int e^{-x} \cos(7x) \, dx \) using integration by parts, we will need to apply the method twice. Here are the steps:

### Step 1: Set up integration by parts
Recall the integration by parts formula:

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

We choose:
- \( u = \cos(7x) \), so \( du = -7 \sin(7x) \, dx \)
- \( dv = e^{-x} \, dx \), so \( v = -e^{-x} \)

### Step 2: Apply integration by parts
Now apply the formula:

\[
\int e^{-x} \cos(7x) \, dx = -e^{-x} \cos(7x) - \int -e^{-x} (-7 \sin(7x)) \, dx
\]

Simplifying:

\[
\int e^{-x} \cos(7x) \, dx = -e^{-x} \cos(7x) + 7 \int e^{-x} \sin(7x) \, dx
\]

### Step 3: Use integration by parts again
Now we need to solve the integral \( \int e^{-x} \sin(7x) \, dx \). We apply integration by parts again, this time choosing:
- \( u = \sin(7x) \), so \( du = 7 \cos(7x) \, dx \)
- \( dv = e^{-x} \, dx \), so \( v = -e^{-x} \)

Applying the formula again:

\[
\int e^{-x} \sin(7x) \, dx = -e^{-x} \sin(7x) - \int -e^{-x} (7 \cos(7x)) \, dx
\]

Simplifying:

\[
\int e^{-x} \sin(7x) \, dx = -e^{-x} \sin(7x) - 7 \int e^{-x} \cos(7x) \, dx
\]

### Step 4: Solve the system of equations
Now we have:

\[
I = \int e^{-x} \cos(7x) \, dx
\]
and
\[
\int e^{-x} \sin(7x) \, dx = -e^{-x} \sin(7x) - 7I
\]

Substitute this back into the equation for the original integral:

\[
I = -e^{-x} \cos(7x) + 7(-e^{-x} \sin(7x) - 7I)
\]

Simplify:

\[
I = -e^{-x} \cos(7x) - 7e^{-x} \sin(7x) - 49I
\]

### Step 5: Solve for \( I \)
Bring like terms together:

\[
I + 49I = -e^{-x} (\cos(7x) + 7 \sin(7x))
\]

\[
50I = -e^{-x} (\cos(7x) + 7 \sin(7x))
\]

Finally, solve for \( I \):

\[
I = \frac{-e^{-x} (\cos(7x) + 7 \sin(7x))}{50}
\]

Thus, the solution to the integral is:

\[
\int e^{-x} \cos(7x) \, dx = \frac{-e^{-x} (\cos(7x) + 7 \sin(7x))}{50} + C
\]

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

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Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. How would you integrate \( \int e^{2x} \sin(5x) \, dx \)?
2. Can you apply integration by parts to \( \int x e^{-x} \, dx \)?
3. How would you solve \( \int e^{-3x} \cos(4x) \, dx \)?
4. What is the integral of \( \int e^x \cos(2x) \, dx \)?
5. How does the integration by parts formula relate to the product rule for differentiation?

**Tip**: Always check if repeating integration by parts will simplify the problem, especially with functions that involve products of exponential and trigonometric terms.

Learn how to solve the integral ∫ e^-x * cos(7x) dx using integration by parts. This detailed step-by-step solution explains how to apply the integration by parts method twice to solve the problem. Ideal for students at the undergraduate level studying calculus.

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