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.

---

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.