Answer:

Explanation:
We would like to integrate the following integral:

Since this is a product of two functions, we can consider using Integration by Parts given by:

So, let’s choose our u and dv. We can choose u base on the following guidelines: LIATE; or, logarithmic, inverse trig., algebraic, trigonometric, and exponential.
Since trigonometric comes before exponential, we will let:

By finding the differential of the left and integrating the right, we acquire:

So, our integral becomes:

Simplify:

Since we ended up with another integral of a product of two functions, we can apply integration by parts again. Using the above guidelines, we get that:

By finding the differential of the left and integrating the right, we acquire:

This yields:
![\displaystyle \int e^(-x)\cdot \cos(2x)\, dx=-e^(-x)\cos(2x)-2\Big[(\sin(2x))(-e^(-x))-\int (-e^(-x))(2\sin(2x))\, dx\Big]](https://img.qammunity.org/2021/formulas/mathematics/college/a8p76zad3umejv5tj12itggf547o005f8k.png)
Simplify:
![\displaystyle \int e^(-x)\cdot \cos(2x)\, dx=-e^(-x)\cos(2x)-2\Big[-e^(-x)\sin(2x)+2\int e^(-x)\cos(2x)\, dx\Big]](https://img.qammunity.org/2021/formulas/mathematics/college/udp2l9ex1svw0wmnbljjqrmwr8zw0g54b2.png)
We can distribute:

The integral on the right is the same as our original integral. So, we can isolate it:

Combine like integrals:

We can factor out an e⁻ˣ from the right:

Dividing both sides by 5 yields:

Rewrite. We of course also need the constant of integration. Therefore, our final answer is:
