1. **Stating the problem:** We want to evaluate the integral $$\int_0^\infty e^{-ax} \sin(\beta x) \, dx$$ where $a > 0$ and $\beta$ are constants. 2. **Formula and important rules:** This is a standard integral involving an exponential decay multiplied by a sine function. The formula for such integrals is: $$\int_0^\infty e^{-px} \sin(qx) \, dx = \frac{q}{p^2 + q^2}$$ where $p > 0$. 3. **Applying the formula:** Here, $p = a$ and $q = \beta$. Since $a > 0$, the formula applies directly. 4. **Final answer:** $$\int_0^\infty e^{-ax} \sin(\beta x) \, dx = \frac{\beta}{a^2 + \beta^2}$$ This result shows how the integral converges due to the exponential decay and oscillates due to the sine function.