1) \( \int \operatorname{cosa}^{2} x \cdot d x= \)

To solve the integral \( \int \cos^2(x) \, dx \), you can use the identity that relates cosine squared to a double angle: \[ \cos^2(x) = \frac{1 + \cos(2x)}{2}. \] Substituting this into the integral gives: \[ \int \cos^2(x) \, dx = \int \frac{1 + \cos(2x)}{2} \, dx = \frac{1}{2} \int (1 + \cos(2x)) \, dx. \] Now integrate term by term: \[ = \frac{1}{2} \left( x + \frac{1}{2} \sin(2x) \right) + C = \frac{x}{2} + \frac{\sin(2x)}{4} + C, \] where \( C \) is the constant of integration. So, \( \int \cos^2(x) \, dx = \frac{x}{2} + \frac{\sin(2x)}{4} + C \). For a fun fact: Did you know that integrals like these pop up in various fields such as physics and engineering? They often help determine areas, volumes, and even the behavior of waves! Also, if you want to dive deeper into the world of integrals, consider exploring resources like "Calculus Made Easy" by Silvanus P. Thompson. This book simplifies complex calculus concepts and makes learning integrals fun and approachable!