integrate by parts e^-x*cos(7x)dx using integration by parts and reduction formula

To solve the integral of \( e^{-x} \cos(7x) \, dx \) using **integration by parts** and a **reduction formula**, we proceed step by step.

### Step 1: Set up the integral and use integration by parts

We need to evaluate the integral:

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

Using the integration by parts formula:

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

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

### Step 2: Apply integration by parts

Using the formula:

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

Let the new integral be:

\[
I_1 = \int e^{-x} \sin(7x) \, dx
\]

### Step 3: Use integration by parts again on \( I_1 \)

We apply integration by parts again to \( I_1 \).

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

Applying integration by parts:

\[
I_1 = -e^{-x} \sin(7x) - 7 \int e^{-x} \cos(7x) \, dx
\]

### Step 4: Solve the system of equations

We now have two expressions for our integrals:

1. \( I = -e^{-x} \cos(7x) + 7 I_1 \)
2. \( I_1 = -e^{-x} \sin(7x) - 7 I \)

Substitute the expression for \( I_1 \) from equation 2 into equation 1:

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

Simplifying:

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

Now, collect all terms involving \( I \) on one side:

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

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

### Step 5: Solve for \( I \)

Finally, divide both sides by 50:

\[
I = -\frac{1}{50} e^{-x} (\cos(7x) + 7 \sin(7x)) + C
\]

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

### Final Answer:

\[
\int e^{-x} \cos(7x) \, dx = -\frac{1}{50} e^{-x} (\cos(7x) + 7 \sin(7x)) + C
\]

Would you like more details, or do you have any questions?

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### Related Questions:
1. How do we use integration by parts in other trigonometric-exponential integrals?
2. What is the general reduction formula for integrals involving \( e^{ax} \cos(bx) \)?
3. How can the technique be extended to \( \int e^{ax} \sin(bx) \, dx \)?
4. Can we apply a similar approach to integrals with higher powers of \( x \) in the exponent?
5. How do we solve integrals involving products of two trigonometric functions?

**Tip**: In complex integrals, always look for patterns that allow you to simplify the system or use symmetry.

Integrate by parts e^-x*cos(7x)dx using integration by parts and reduction formula

Learn how to integrate e^-x * cos(7x) using integration by parts and a reduction formula. This step-by-step guide explains the key formulas and methods for solving complex integrals involving exponential and trigonometric functions. Suitable for advanced calculus students.

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