1. **Problem statement:** Evaluate the integral $$\int_0^\infty \frac{e^{at}}{(1+t^2)^2} \, dt$$ using Feynman's technique (differentiation under the integral sign). 2. **Recall Feynman's technique:** If we have an integral depending on a parameter $a$, say $$I(a) = \int f(t,a) \, dt,$$ then differentiating under the integral sign gives $$I'(a) = \int \frac{\partial}{\partial a} f(t,a) \, dt.$$ This can simplify the integral. 3. **Define the integral:** Let $$I(a) = \int_0^\infty \frac{e^{at}}{(1+t^2)^2} \, dt.$$ 4. **Differentiate $I(a)$ with respect to $a$:** $$I'(a) = \int_0^\infty \frac{\partial}{\partial a} \left( \frac{e^{at}}{(1+t^2)^2} \right) dt = \int_0^\infty \frac{t e^{at}}{(1+t^2)^2} \, dt.$$ 5. **Rewrite $I'(a)$:** We want to express $I'(a)$ in a form that can be integrated or related to known integrals. 6. **Use integration by parts or known integrals:** Consider the integral $$J(a) = \int_0^\infty \frac{e^{at}}{1+t^2} \, dt.$$ Differentiating $J(a)$ with respect to $a$: $$J'(a) = \int_0^\infty \frac{t e^{at}}{1+t^2} \, dt.$$ 7. **Relate $I'(a)$ and $J'(a)$:** Note that $$I'(a) = \int_0^\infty \frac{t e^{at}}{(1+t^2)^2} \, dt = -\frac{1}{2} \frac{d}{da} \int_0^\infty \frac{e^{at}}{1+t^2} \, dt = -\frac{1}{2} J'(a).$$ 8. **Therefore,** $$I'(a) = -\frac{1}{2} J'(a) \implies I(a) = -\frac{1}{2} J(a) + C,$$ where $C$ is a constant. 9. **Evaluate $J(a)$:** For $a<0$, the integral $$J(a) = \int_0^\infty \frac{e^{at}}{1+t^2} \, dt = \frac{\pi}{2} e^{-|a|}$$ (known integral formula). 10. **Determine constant $C$:** As $a \to -\infty$, $I(a) \to 0$ and $J(a) \to 0$, so $$0 = -\frac{1}{2} \cdot 0 + C \implies C=0.$$ 11. **Final answer:** $$I(a) = -\frac{1}{2} J(a) = -\frac{1}{2} \cdot \frac{\pi}{2} e^{-|a|} = -\frac{\pi}{4} e^{-|a|}.$$