Integral of \( \displaystyle \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} \)
Problem 4.234 · easy
Find \( \displaystyle \int \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}\, dx \]integral rewrite substitutionStart with the integral of the given function. Rewrite the fraction as a product. Recognize that cos(x) is the derivative of sin(x).✓ Proved
- \[ = \ln{\left(\sin{\left(x \right)} \right)} \]antiderivativeThe integral of 1/u du is log(u).✓ Proved
Answer \( \ln{\left(\sin{\left(x \right)} \right)} + C \)
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 sin(x) = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(x) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(x) = 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 (style) — Step 3 is labeled 'substitution' but performs a recognition of the derivative (a 'derivative' or 'rewrite' step); the actual substitution is performed in Step 4. Step 3 should be labeled 'derivative' or 'rewrite' to accurately reflect that it identifies cos(x) as diff(sin(x), x).
Every verdict on record (4)
qwen3.6:27b-mlx: fail (style) 2026-10-03 — Step 3 is labeled 'substitution' but performs a recognition of the derivative (a 'derivative' or 'rewrite' step); the actual substitution is performed in Step 4. Step 3 should be labeled 'derivative' or 'rewrite' to accurately reflect that it identifies cos(x) as diff(sin(x), x).gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 3 is labeled 'substitution' but performs a derivative recognition (rewriting cos(x) as diff(sin(x), x)), which is not a substitution step. Step 4 is labeled 'antiderivative' but performs the integration (substitution) itself; the label should be 'substitution' or 'integral'.gpt-oss:20b: pass 2026-10-03
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 2026-10-03 with SymPy 1.14.0.