Integral of \( \displaystyle \tan^{2}{\left(2 x - 1 \right)} \)
Problem 4.258 · medium
Find \( \displaystyle \int \tan^{2}{\left(2 x - 1 \right)} \, dx \). (Omit the constant of integration.)
- \[ \int \tan^{2}{\left(2 x - 1 \right)}\, dx \]integral trig-identityStart with the integral of the given function. Use the identity tan(u)**2 = sec(u)**2 - 1.✓ Proved
- \[ = - \int 1\, dx + \int \sec^{2}{\left(2 x - 1 \right)}\, dx \]algebraSimplify the expression by combining the integrals.Not checked
- \[ = - x + \int \sec^{2}{\left(2 x - 1 \right)}\, dx \]antiderivativeEvaluate the integral of the constant 1.✓ Proved
- \[ = - x + \frac{\tan{\left(2 x - 1 \right)}}{2} \]antiderivativeEvaluate the integral of sec(2*x - 1)**2 using substitution.Not checked
Answer \( - x + \frac{\tan{\left(2 x - 1 \right)}}{2} + C \)
Lines: 4 proved, 2 not checked. 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 tan has poles at odd multiples of pi/2 |
| 3 | Not checked | — | simplify left tan(2*x - 1)/2 - Integral(sec(2*x - 1)**2, x); no point in the sample was defined on both lines tan has poles at odd multiples of pi/2 sec has poles at odd multiples of pi/2 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 sec has poles at odd multiples of pi/2 |
| 5 | Not checked | — | simplify left -tan(2*x - 1)/2 + Integral(sec(2*x - 1)**2, x); no point in the sample was defined on both lines sec has poles at odd multiples of pi/2 tan has poles at odd multiples of pi/2 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 tan has poles at odd multiples of pi/2 |
| 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) — Step 2 is labeled 'trig-identity' but does not apply the identity; it merely adds and subtracts Integral(1, x). The actual application of tan^2 = sec^2 - 1 is missing or conflated with the algebraic manipulation in Step 3, violating the one-rule-per-step constraint.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — Step 2 is labeled 'trig-identity' but does not apply the identity; it merely adds and subtracts Integral(1, x). The actual application of tan^2 = sec^2 - 1 is missing or conflated with the algebraic manipulation in Step 3, violating the one-rule-per-step constraint.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (error) 2026-10-04 — Step 2 is labeled 'trig-identity' but does not apply the identity; it merely adds and subtracts Integral(1, x). The actual application of tan^2 = sec^2 - 1 happens in Step 3, which is incorrectly labeled 'algebra'. Furthermore, Step 3 combines two operations (applying the identity and splitting the integral), violating the one-rule-per-step constraint.gpt-oss:20b: pass 2026-10-04
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 2026-10-04 with SymPy 1.14.0.