Integral of \( \displaystyle \frac{e^{2 x}}{e^{2 x} + 1} \)
Problem 4.659 · medium
Find \( \displaystyle \int \frac{e^{2 x}}{e^{2 x} + 1} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{e^{2 x}}{e^{2 x} + 1}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \frac{2 e^{2 x}}{2 e^{2 x} + 2}\, dx \]rewrite algebraMultiply and divide by 2 to prepare for substitution. Distribute the 2 in the denominator.✓ Proved
- \[ = \int \frac{e^{2 x}}{e^{2 x} + 1}\, dx \]simplifySimplify the fraction.✓ Proved
- \[ = \frac{\ln{\left(e^{2 x} + 1 \right)}}{2} \]antiderivativeApply the antiderivative rule for the substitution u = exp(2*x) + 1.✓ Proved
Answer \( \frac{\ln{\left(e^{2 x} + 1 \right)}}{2} + C \)
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(2*x) + 1 = 0 undefined where 2*exp(2*x) + 2 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*exp(2*x) + 2 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*exp(2*x) + 2 = 0 undefined where exp(2*x) + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where exp(2*x) + 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: fail (error) — Steps 2, 3, and 4 perform a circular manipulation that returns the expression to its original form, making Step 5 an unjustified jump to the answer without showing the actual substitution or integration logic. Specifically, Step 5 claims to apply an antiderivative rule but skips the necessary change of variables or recognition of the derivative of the denominator.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: fail (error) 2026-10-08 — Steps 2, 3, and 4 perform a circular manipulation that returns the expression to its original form, making Step 5 an unjustified jump to the answer without showing the actual substitution or integration logic. Specifically, Step 5 claims to apply an antiderivative rule but skips the necessary change of variables or recognition of the derivative of the denominator.gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: fail (error) 2026-10-08 — Steps 2, 3, and 4 are redundant and logically circular; Step 2 rewrites the expression into an equivalent form, Step 3 expands it, and Step 4 immediately reverts it to the original expression in Step 1. Furthermore, Step 5 claims to apply an antiderivative rule but skips the necessary substitution step (u = exp(2*x) + 1) entirely, jumping from the integral form to the final answer without showing the change of variables or the resulting integral in terms of u.
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 2026-10-08 with SymPy 1.14.0.