putnam_like Set_2/B6: stated integral identity is false (cos should be sin); rubric contains a compensating algebra slip

## Summary

The Putnam-like question `putnam_like/Set_2/B6` asks to show

$$\int_0^{\infty}\frac{e^{-2x}}{1+x^2}\,dx=\int_0^{\infty}\frac{\cos x}{2+x}\,dx,$$

but this identity is **false**. The intended statement almost certainly has $\sin x$ in place of $\cos x$.

## Why the stated identity is false

Writing $\frac{1}{2+x}=\int_0^\infty e^{-(2+x)u}\,du$ and exchanging the order of integration gives the Laplace pairings

$$\int_0^{\infty}\frac{\cos x}{2+x}\,dx=\int_0^{\infty}\frac{u\,e^{-2u}}{1+u^2}\,du,
\qquad
\int_0^{\infty}\frac{\sin x}{2+x}\,dx=\int_0^{\infty}\frac{e^{-2u}}{1+u^2}\,du.$$

So it is the **sine** integral that equals the right-hand side. Numerically (chunked Simpson integration over $[k\pi,(k+1)\pi]$ out to $T=4\cdot 10^5$, agreeing with the Laplace forms to 4+ decimal places):

$$\int_0^{\infty}\frac{e^{-2x}}{1+x^2}\,dx \approx 0.39902 \approx \int_0^{\infty}\frac{\sin x}{2+x}\,dx,
\qquad
\int_0^{\infty}\frac{\cos x}{2+x}\,dx \approx 0.14454.$$

## Where the grading scheme hides the error

The rubric (`Set_2/B6/rubrics/grading_scheme.md`) derives $f''(t)=\tfrac1t-f(t)$ for $f(t)=\int_0^\infty \tfrac{e^{-tx}}{1+x^2}\,dx$ (correct), and then computes for $g(t)=\int_0^\infty\tfrac{\cos x}{t+x}\,dx$:

$$g''(t)=2\int_0^{\infty}\frac{\cos x}{(t+x)^3}\,dx=\left.-\frac{\cos x}{(t+x)^2}\right|_0^{\infty}-\int_0^{\infty}\frac{\sin x}{(t+x)^2}\,dx=\frac{1}{t}-\dots$$

The boundary term is $\cos 0/(t+0)^2=\mathbf{1/t^2}$, not $1/t$. With the correct boundary term, $g''(t)=\tfrac{1}{t^2}-g(t)$ — a *different* ODE — and the "same ODE plus decay at infinity ⟹ equal" argument collapses, consistent with the numerics above.

For $h(t)=\int_0^\infty\frac{\sin x}{t+x}\,dx$ the same computation genuinely gives $h''(t)=\tfrac1t-h(t)$ with $h(t)\to 0$, so the rubric's ODE argument works verbatim for the sine version. This strongly suggests the problem was authored with $\sin$, and the $\cos$ is a transcription slip that the compensating $1/t^2 \to 1/t$ slip in the rubric made invisible during review.

## Impact

- Models evaluated on this item were asked to "show" a false identity; the rubric awards points for reproducing the flawed derivation, while a model that correctly observes the identity is false (and repairs it to the sine version) would be penalized.
- The companion Lean repository inherited the same statement; a fix is proposed in google-deepmind/formal-putnam-like#7 (integrand `cos → sin` in `PutnamLike/Set2/B6.lean`, where an earlier commit had already corrected the integration range but kept `cos`).

Since the released `samples/` were graded against the published question text, an erratum note may be more appropriate than editing `question.md` in place — filing this issue so the record exists either way.

Read the published question.md and putnam_like/Set_2/B6/rubrics/grading_scheme.md, then compare them with PutnamLike/Set2/B6.lean in the companion repository. Verify the integral identity and rubric derivation, and determine whether the correction should be an erratum or edits to the released materials; done means the question, grading scheme, and evaluation record no longer assert incompatible mathematics.

Written by the indexing model from the issue text.