How to Solve the Integral of e^x cos(2x) Using Integration By Parts

Integrating functions involving exponential and trigonometric elements like $ \int e^x \cos(2x) \, dx $ can pose significant challenges. This integral is particularly interesting because it combines growth and oscillation, which is typical in physics and engineering contexts. Here, we'll explore the solution using the method of integration by parts.

Integration by Parts Formula

Integration by parts is derived from the product rule in differentiation and is articulated as:

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

For our integral, we select:

$ u = e^x $, which simplifies $ du = e^x \, dx $.
$ dv = \cos(2x) \, dx $, leading to $ v = \frac{1}{2} \sin(2x) $ after integration.

First Application of Integration by Parts

The first application of the formula simplifies our integral to:

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

Reapplying Integration by Parts

Applying integration by parts again to the second integral, we set up a new pair of $ u $ and $ dv $:

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

Solving the System of Equations

The simultaneous equations derived from these steps give us the expression:

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

Conclusion

This integration showcases the utility of the integration by parts technique, proving effective for complex integrals involving products of exponential and trigonometric functions. Such integrals frequently arise in various scientific and engineering fields, emphasizing the importance of mastering these techniques.