Manifolds Unfolded

Feynman’s Trick, Leibniz’s Theorem, and a Hidden Simplicity

Some integrals look innocent yet refuse every standard technique we throw at them. One such example is

The logarithm in the denominator immediately blocks substitution, integration by parts, and elementary transformations. Yet, with a single conceptual shift—an idea popularized by Richard Feynman—this integral collapses into a closed-form expression with almost no effort.

This post unpacks that idea carefully, rigorously, and elegantly.

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The Key Insight: Introduce a Parameter

Instead of attacking the integral head-on, embed it into a one-parameter family:

The original problem corresponds to the special value $I(5)$. The advantage of this formulation is subtle but powerful: while $I(t)$ itself looks complicated, its derivative with respect to $t$ is remarkably simple.

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Differentiation Under the Integral Sign

We now differentiate $I(t)$ with respect to $t$. Formally,

Interchanging differentiation and integration (justified by Leibnitz theorem),

This integral is elementary:

Thus,

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Reconstructing the Integral

Integrating with respect to $t$,

where $C$ is a constant.

To determine $C$, evaluate $I(t)$ at $t = 0$:

This immediately gives $C = 0$, and therefore,

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The Final Answer

Substituting $t = 5$,

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Why This Method Is So Powerful

This example showcases one of the most beautiful ideas in analysis: changing the problem instead of fighting it. By introducing a parameter and differentiating with respect to it, a non-elementary integral is reduced to a trivial computation.

What appears at first as a technical trick is, in fact, a profound principle—one that reappears across analysis, geometry, quantum mechanics, and statistical physics.

On Manifolds Unfolded, this philosophy lies at the heart of mathematics: clarity emerges not from brute force, but from the right viewpoint.