(1) \( \int \sin (\omega t+\varphi) \mathrm{d} t(\omega, \varphi \quad \) are constants \( ) \)

To integrate \( \sin (\omega t + \varphi) \), we can use a simple substitution method. The integral becomes: \[ \int \sin(\omega t + \varphi) \, \mathrm{d}t = -\frac{1}{\omega} \cos(\omega t + \varphi) + C \] where \( C \) is the constant of integration. This means you'll get a negative cosine function scaled by the constant \( \frac{1}{\omega} \). This integration is widely used in physics and engineering, especially in oscillatory motion problems like analyzing waves, pendulums, and other periodic phenomena. Understanding this integration can help in modeling real-world systems, such as electrical circuits and mechanical vibrations, making it a valuable tool for students and professionals alike!