Integral of \( \displaystyle \frac{e^{x - 1}}{e^{x - 1} + 1} \)
Problem 4.733 · medium
Find \( \displaystyle \int \frac{e^{x - 1}}{e^{x - 1} + 1} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{e^{x - 1}}{e^{x - 1} + 1}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \frac{e^{x}}{e \left(\frac{e^{x}}{e} + 1\right)}\, dx \]algebraRewrite the exponent using properties of exponents.≈ Checked numerically
- \[ = \int \frac{e^{x}}{e^{x} + e}\, dx \]simplifyMultiply the numerator and denominator by exp(1) to simplify.✓ Proved
- \[ = \int \frac{1}{e^{1 - x} + 1}\, dx \]rewriteDivide numerator and denominator by exp(x).≈ Checked numerically
- \[ = \ln{\left(e^{x} + e \right)} \]antiderivativeThe antiderivative is found using substitution or recognizing the derivative of the denominator.≈ Checked numerically
Answer \( \ln{\left(e^{x - 1} + 1 \right)} + C \)
Lines: 2 proved, 4 checked numerically. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left log((exp(x - 1) + 1)/(exp(x) + E)); numeric agreement only, at 24 of 24 sampled points undefined where exp(x - 1) + 1 = 0 undefined where exp(-1)*exp(x) + 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where exp(-1)*exp(x) + 1 = 0 undefined where exp(x) + E = 0 |
| 4 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left log((exp(x) + E)/(exp(x - 1) + 1)) - 1; numeric agreement only, at 24 of 24 sampled points undefined where exp(x) + E = 0 undefined where exp(1 - x) + 1 = 0 |
| 5 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left log((exp(x - 1) + 1)/(exp(x) + E)) + 1; numeric agreement only, at 24 of 24 sampled points undefined where exp(1 - x) + 1 = 0 log is undefined for non-positive arguments |
| answer | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: final line against the stated answer: simplify left log((exp(x) + E)/(exp(x - 1) + 1)); numeric agreement only, at 24 of 24 sampled points log is undefined for non-positive arguments |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy differentiated the stated antiderivative back to the integrand |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — Step 4 violates the 'one change per step' rule by simultaneously dividing the numerator and denominator by exp(x) and rewriting the resulting expression, which is a complex algebraic manipulation not covered by a single 'rewrite' label. Furthermore, Step 5 jumps from the integral of 1/(1 + exp(1-x)) to log(exp(x) + exp(1)) without showing the substitution or integration steps, effectively applying multiple rules (substitution and integration) at once.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: fail (error) 2026-10-09 — Step 4 violates the 'one change per step' rule by simultaneously dividing the numerator and denominator by exp(x) and rewriting the resulting expression, which is a complex algebraic manipulation not covered by a single 'rewrite' label. Furthermore, Step 5 jumps from the integral of 1/(1 + exp(1-x)) to log(exp(x) + exp(1)) without showing the substitution or integration steps, effectively applying multiple rules (substitution and integration) at once.gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: fail (error) 2026-10-09 — Step 5 claims the antiderivative of 1/(1 + exp(1 - x)) is log(exp(x) + exp(1)). Differentiating the result gives exp(x)/(exp(x) + exp(1)), which matches the integrand in Step 3, but the integrand in Step 4 is 1/(1 + exp(1 - x)). While these two integrands are algebraically equivalent, the step label 'antiderivative' implies a direct recognition or standard form application. However, the transition from Step 4 to Step 5 skips the necessary algebraic simplification back to a recognizable form (like Step 3) or explicitly performing the substitution. More critically, the note says 'using substitution or recognizing...', but the step itself jumps from a rewritten form (Step 4) to the final answer without showing the substitution variable change or the algebraic equivalence to the form in Step 3. If we treat Step 4 as the current state, applying 'antiderivative' directly is misleading because 1/(1+e^(1-x)) is not a standard form usually memorized; it requires rewriting to e^x/(e^x + e) first. The step effectively combines 'rewrite' (to get back to Step 3's form) and 'antiderivative'. This violates the 'one rule per step' constraint.
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
gemma4:26b, checked 2026-10-09 with SymPy 1.14.0.