Integral of \( \displaystyle \frac{e^{3 x + 1}}{e^{3 x + 1} + 1} \)
Problem 4.1241 · medium Video
Find \( \displaystyle \int \frac{e^{3 x + 1}}{e^{3 x + 1} + 1} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{e^{3 x + 1}}{e^{3 x + 1} + 1}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \frac{3 e^{3 x + 1}}{3 e^{3 x + 1} + 3}\, dx \]rewrite algebraRewrite the numerator to reveal the derivative of the inner function. Simplify the constants in the integrand.✓ Proved
- \[ = \int \frac{e^{3 x + 1}}{e^{3 x + 1} + 1}\, dx \]simplify substitution algebra simplifySimplify the expression. This step is not needed if we use the chain rule directly, but we show the structure. Simplify the expression. Simplify the integrand.✓ Proved
- \[ = \frac{\ln{\left(e^{3 x + 1} + 1 \right)}}{3} \]antiderivativeEvaluate the integral using the substitution u = exp(3*x + 1) + 1.✓ Proved
Answer \( \frac{\ln{\left(e^{3 x + 1} + 1 \right)}}{3} + C \)
Every line of this solution was proved by the computer algebra system SymPy. 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. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where exp(3*x + 1) + 1 = 0 undefined where 3*exp(3*x + 1) + 3 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*exp(3*x + 1) + 3 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*exp(3*x + 1) + 3 = 0 undefined where exp(3*x + 1) + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where exp(3*x + 1) + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where exp(3*x + 1) + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where exp(3*x + 1) + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where exp(3*x + 1) + 1 = 0 log is undefined for non-positive arguments |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 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: fail (style) — Step 5 is labeled "substitution" but only performs an algebraic rewrite; no substitution is actually applied. The step is unnecessary and mislabels the rule used.qwen3.6:27b-mlx: fail (error) — Steps 2-4 and 5-7 are circular and redundant, effectively rewriting the integrand to itself without progress. Step 5 is mathematically nonsensical as a substitution step, introducing a complex fraction that simplifies back to the original integrand in Step 7, failing to actually perform the substitution u = exp(3*x + 1) + 1.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-19 — Steps 2-4 and 5-7 are circular and redundant, effectively rewriting the integrand to itself without progress. Step 5 is mathematically nonsensical as a substitution step, introducing a complex fraction that simplifies back to the original integrand in Step 7, failing to actually perform the substitution u = exp(3*x + 1) + 1.gpt-oss:20b: fail (style) 2026-10-19 — Step 5 is labeled "substitution" but only performs an algebraic rewrite; no substitution is actually applied. The step is unnecessary and mislabels the rule used.qwen3.6:27b-mlx: fail (error) 2026-10-16 — Steps 2 through 7 are logically incoherent and mathematically incorrect. Step 2 introduces a factor of 3 in the numerator and denominator that does not simplify to the original expression in a meaningful way for substitution, and subsequent steps (5-7) perform nonsensical algebraic manipulations (multiplying by 1 in a convoluted way) that do not constitute a valid substitution step. The transition from Step 7 to Step 8 claims to use substitution but skips the actual integration logic, effectively jumping to the answer without showing the correct $u$-substitution process (where $du = 3e^{3x+1}dx$).gpt-oss:20b: fail (error) 2026-10-16 — Step 5 applies multiple manipulations at once and introduces an incorrect factorization; it neither follows a single allowed rule nor preserves the integrand, leading to a logically flawed transition.qwen3.6:27b-mlx: fail (error) 2026-10-16 — Steps 2-7 are logically incoherent and redundant; specifically, Step 5 introduces a nonsensical expression that does not simplify to the integrand in Step 6, and the transition from Step 7 to Step 8 skips the actual integration logic while claiming a substitution was performed in a previous invalid step.gpt-oss:20b: fail (misleading) 2026-10-16 — Step 5 applies multiple transformations at once and introduces an incorrect factorization of the integrand; it misleads the reader about how the substitution should be performed. The subsequent simplifications in steps 6–7 are also inconsistent with a single-rule application and do not correctly reflect the substitution u=exp(3*x+1)+1. The final antiderivative is correct, but the intermediate steps are misleading and violate the one‑rule‑per‑step rule.
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 2026-10-19 with SymPy 1.14.0.