What is the definite integral of cos^2(x) using integration by parts?

To determine the definite integral of cos²(x), we can utilize the identity for cos²(x) that simplifies the integration process. The identity states that cos²(x) can be expressed as:

[

\cos^2(x) = \frac{1 + \cos(2x)}{2}

]

Using this identity, the integral becomes:

[

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

]

Breaking this down, we can integrate each part separately:

1. The integral of 1 is simply x.

2. The integral of cos(2x) can be solved using basic integration rules, yielding (\frac{1}{2}\sin(2x)).

Putting this together:

[

\int \cos^2(x) , dx = \frac{1}{2} \int 1 , dx + \frac{1}{2} \int \cos(2x) , dx = \frac{x}{2} + \frac{1}{2} \cdot \frac{1}{2}\sin(2x) + C = \frac{x