Integral of \( \displaystyle \frac{e^{3 x - 1}}{e^{3 x - 1} + 1} \)
Problem 4.466 · medium
Find \( \displaystyle \int \frac{e^{3 x - 1}}{e^{3 x - 1} + 1} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{e^{3 x - 1}}{e^{3 x - 1} + 1}\, dx \]integral algebraStart with the integral of the given function. Rewrite the numerator to facilitate substitution.✓ Proved
- \[ = \int \frac{3 e^{3 x - 1}}{3 e^{3 x - 1} + 3}\, dx \]algebraDistribute the constant 3.✓ Proved
- \[ = \int \frac{e^{3 x - 1}}{e^{3 x - 1} + 1}\, dx \]algebra linearityRearrange terms. Pull out the constant factor.✓ Proved
- \[ = \frac{\int \frac{3 e^{3 x - 1}}{e^{3 x - 1} + 1}\, dx}{3} \]linearityApply linearity of the integral.✓ Proved
- \[ = \frac{\ln{\left(e^{3 x - 1} + 1 \right)}}{3} \]antiderivativeThe integral is of the form u'/u, which results in log(u).✓ 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.
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 |
| 3 | ✓ 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 |
| 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 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: passqwen3.6:27b-mlx: pass — The solution correctly manipulates the integrand to identify the form u'/u and applies the antiderivative rule. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly manipulates the integrand to identify the form u'/u and applies the antiderivative rule. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.gpt-oss:20b: fail (error) 2026-10-06 — Step 3 applies two algebraic changes at once (moving the factor 3 from the numerator to the denominator). Each step must alter only one element; this step violates that rule.qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly manipulates the integrand to match the form u'/u and applies the antiderivative rule. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
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 2026-10-06 with SymPy 1.14.0.