We are asked to use integration by parts to solve the integral

$I = \int e^x \cos(2x) \, dx$

Step 1: Set up integration by parts

Recall the formula for integration by parts:

$\int u \, dv = uv - \int v \, du$

We need to select parts for $u$ and $dv$. Let:

• $u = e^x$, so $du = e^x \, dx$
• $dv = \cos(2x) \, dx$, so $v = \frac{1}{2} \sin(2x)$ (since the integral of $\cos(2x)$ is $\frac{1}{2} \sin(2x)$)

Step 2: Apply integration by parts

Using the formula for integration by parts:

$I = e^x \cdot \frac{1}{2} \sin(2x) - \int \frac{1}{2} \sin(2x) \cdot e^x \, dx$

This simplifies to:

$I = \frac{e^x}{2} \sin(2x) - \frac{1}{2} \int e^x \sin(2x) \, dx$

Step 3: Solve the new integral

We now need to compute $\int e^x \sin(2x) \, dx$. Let this integral be $J$, so:

$J = \int e^x \sin(2x) \, dx$

We apply integration by parts again. Let:

• $u = e^x$, so $du = e^x \, dx$
• $dv = \sin(2x) \, dx$, so $v = -\frac{1}{2} \cos(2x)$

Thus:

$J = e^x \cdot \left( -\frac{1}{2} \cos(2x) \right) - \int -\frac{1}{2} \cos(2x) \cdot e^x \, dx$

Simplifying:

$J = -\frac{e^x}{2} \cos(2x) + \frac{1}{2} \int e^x \cos(2x) \, dx$

Step 4: Solve for $I$

Notice that the integral on the right side of the equation for $J$ is the original integral $I$. Thus:

$J = -\frac{e^x}{2} \cos(2x) + \frac{1}{2} I$

Substitute this into the equation for $I$:

$I = \frac{e^x}{2} \sin(2x) - \frac{1}{2} \left( -\frac{e^x}{2} \cos(2x) + \frac{1}{2} I \right)$

Simplify:

$I = \frac{e^x}{2} \sin(2x) + \frac{e^x}{4} \cos(2x) - \frac{1}{4} I$

Step 5: Solve for $I$

Now, collect terms involving $I$:

$I + \frac{1}{4} I = \frac{e^x}{2} \sin(2x) + \frac{e^x}{4} \cos(2x)$

This simplifies to:

$\frac{5}{4} I = \frac{e^x}{2} \sin(2x) + \frac{e^x}{4} \cos(2x)$

Multiply both sides by $\frac{4}{5}$ to solve for $I$:

$I = \frac{2e^x}{5} \sin(2x) + \frac{e^x}{5} \cos(2x)$

Thus, we have shown that:

$\int e^x \cos(2x) \, dx = \frac{2e^x}{5} \sin(2x) + \frac{e^x}{5} \cos(2x) + C$

where $C$ is the constant of integration.

Would you like further details on any specific step, or do you have any questions?

Related Questions:

1. How does integration by parts work for products of exponential and trigonometric functions in general?
2. What is the derivation of the integration formula for $\cos(2x)$?
3. Can we use substitution instead of integration by parts for similar integrals?
4. How do the signs in trigonometric integrals like $\cos(2x)$ and $\sin(2x)$ affect the integration process?
5. What are common mistakes made when solving integrals using integration by parts?

Tip: Always double-check which part of the product you are choosing as $u$ and $dv$ to simplify the resulting integral.