Subjects calculus

Integral M Ex E5D3A4

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Integral M Ex E5D3A4


1. **State the problem:** We need to evaluate the integral $$\int \frac{m}{1 e^x} \, dm$$. 2. **Analyze the integral:** The integral is with respect to $m$, and the integrand is $$\frac{m}{1 e^x} = \frac{m}{e^x}$$ since $1 e^x = e^x$. 3. **Treat $e^x$ as a constant:** Since the integration is with respect to $m$, and $x$ is independent of $m$, $e^x$ is a constant. 4. **Rewrite the integral:** $$\int \frac{m}{e^x} \, dm = \frac{1}{e^x} \int m \, dm$$ 5. **Integrate with respect to $m$:** $$\int m \, dm = \frac{m^2}{2} + C$$ 6. **Combine results:** $$\frac{1}{e^x} \cdot \frac{m^2}{2} + C = \frac{m^2}{2 e^x} + C$$ **Final answer:** $$\int \frac{m}{1 e^x} \, dm = \frac{m^2}{2 e^x} + C$$