Integral of \( \displaystyle \frac{\cos{\left(3 x \right)}}{\sin{\left(3 x \right)}} \)
Problem 4.369 · medium
Find \( \displaystyle \int \frac{\cos{\left(3 x \right)}}{\sin{\left(3 x \right)}} \, dx \). (Omit the constant of integration.)
- \[ \int \frac{\cos{\left(3 x \right)}}{\sin{\left(3 x \right)}}\, dx \]integral algebra rewrite algebra substitution algebraStart with the integral of the given function. Rewrite the integrand by multiplying and dividing by 3. Group the terms to prepare for substitution. Simplify the expression back to a recognizable form. Recognize the derivative of sin(3*x) is 3*cos(3*x). Simplify the constants inside the integral.✓ Proved
- \[ = \frac{\ln{\left(\sin{\left(3 x \right)} \right)}}{3} \]antiderivativeApply the antiderivative rule for 1/u du.✓ Proved
Answer \( \frac{\ln{\left(\sin{\left(3 x \right)} \right)}}{3} + C \)
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 sin(3*x) = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(3*x) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(3*x) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(3*x) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(3*x) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sin(3*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: fail (style) — Step 5 is labeled "substitution" but no substitution is actually performed; the step merely notes that 3*cos(3*x) is the derivative of sin(3*x). No other steps violate the rules.qwen3.6:27b-mlx: fail (error) — Step 6 incorrectly removes the factor of 1/3 from the integrand, which is mathematically invalid and leads to an incorrect final answer (missing the division by 3). Additionally, Step 5 is labeled 'substitution' but does not perform the variable change, and Step 7 applies an antiderivative rule to an expression that does not match the standard form due to the missing constant.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — Step 6 incorrectly removes the factor of 1/3 from the integrand, which is mathematically invalid and leads to an incorrect final answer (missing the division by 3). Additionally, Step 5 is labeled 'substitution' but does not perform the variable change, and Step 7 applies an antiderivative rule to an expression that does not match the standard form due to the missing constant.gpt-oss:20b: fail (style) 2026-10-05 — Step 5 is labeled "substitution" but no substitution is actually performed; the step merely notes that 3*cos(3*x) is the derivative of sin(3*x). No other steps violate the rules.qwen3.6:27b-mlx: fail (error) 2026-10-05 — Step 6 incorrectly removes the factor of 1/3 from the integrand, violating the linearity of integration. The constant 1/3 must remain outside the integral or be carried through, but it disappears entirely in the transition to Step 7.gpt-oss:20b: pass 2026-10-05
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 2026-10-05 with SymPy 1.14.0.