\( \int _ { 0 } ^ { \infty } E d m \quad , \quad E = \sqrt { p ^ { 2 } \cdot c ^ { 2 } + m ^ { 2 } \cdot c ^ { 2 } } \)

Certainly! Let's evaluate the integral you provided: \[ \int_{0}^{\infty} E \, dm \quad \text{where} \quad E = \sqrt{p^2 c^2 + m^2 c^2} \] ### Simplifying the Expression First, factor out \( c \) from the expression for \( E \): \[ E = \sqrt{p^2 c^2 + m^2 c^2} = c \sqrt{p^2 + m^2} \] Therefore, the integral becomes: \[ \int_{0}^{\infty} E \, dm = c \int_{0}^{\infty} \sqrt{p^2 + m^2} \, dm \] ### Evaluating the Integral The integral we need to evaluate is: \[ \int_{0}^{\infty} \sqrt{m^2 + p^2} \, dm \] This is a standard integral in calculus. The antiderivative of \( \sqrt{m^2 + a^2} \) with respect to \( m \) is: \[ \frac{m}{2} \sqrt{m^2 + a^2} + \frac{a^2}{2} \ln\left(m + \sqrt{m^2 + a^2}\right) + C \] Where \( a = p \) in our case. Applying the limits from \( 0 \) to \( \infty \): \[ \left. \left( \frac{m}{2} \sqrt{m^2 + p^2} + \frac{p^2}{2} \ln\left(m + \sqrt{m^2 + p^2}\right) \right) \right|_{0}^{\infty} \] 1. **As \( m \to \infty \):** - \( \frac{m}{2} \sqrt{m^2 + p^2} \approx \frac{m}{2} \cdot m = \frac{m^2}{2} \) → \( \infty \) - \( \frac{p^2}{2} \ln\left(m + \sqrt{m^2 + p^2}\right) \approx \frac{p^2}{2} \ln(2m) \) → \( \infty \) 2. **At \( m = 0 \):** - \( \frac{0}{2} \sqrt{0 + p^2} = 0 \) - \( \frac{p^2}{2} \ln\left(0 + \sqrt{0 + p^2}\right) = \frac{p^2}{2} \ln(p) \) ### Conclusion As \( m \to \infty \), both terms in the antiderivative tend to infinity. Therefore, the definite integral diverges: \[ \int_{0}^{\infty} \sqrt{m^2 + p^2} \, dm = \infty \] Multiplying by \( c \), the original integral also diverges: \[ \int_{0}^{\infty} E \, dm = c \cdot \infty = \infty \] ### Physical Interpretation In a physical context, this divergence suggests that the integral doesn't converge to a finite value under the given conditions. This might imply that integrating energy \( E \) over all possible mass values \( m \) from \( 0 \) to \( \infty \) isn't physically meaningful without additional constraints or considerations. If you have a specific physical scenario or additional constraints in mind, please provide more details, and I can offer a more tailored explanation or solution.