Home›Calculus 1›Definite integrals by substitution›Problem 4.456

Definite integrals by substitution

Problem 4.456 · medium

Evaluate \( \displaystyle \int_{0}^{\frac{\pi}{2}} 3 \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx \).

Let u = sin(x); then du = cos(x) dx, which is in the integrand up to a constant.
Reviewed
\[ \left. \sin{\left(x \right)} \right|_{\substack{ x=0 }} = 0 \]
The lower limit in u.✓ Proved
\[ \left. \sin{\left(x \right)} \right|_{\substack{ x=\frac{\pi}{2} }} = 1 \]
The upper limit in u.✓ Proved
\[ \int\limits_{0}^{1} 3 u^{4}\, du = \frac{3}{5} \]
Integrate in u between the new limits; no back-substitution needed.✓ Proved

Answer \( \frac{3}{5} \)

Lines: 3 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
4	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	numerical quadrature of the original integral

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies u-substitution, properly transforms the limits of integration, and evaluates the resulting definite integral accurately.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies u-substitution, properly transforms the limits of integration, and evaluates the resulting definite integral accurately.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies u-substitution, properly transforms the limits of integration, and evaluates the resulting definite integral accurately.
gpt-oss:20b: fail (misleading) 2026-10-05 — The sentence claims "du = cos(x) dx, which is in the integrand up to a constant." In fact du equals cos(x) dx exactly, with no constant factor. This misleads a student into thinking a constant factor is missing or needed. The rest of the solution is correct, but the wording is inaccurate.

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 generator:structured/definite_substitution, checked 2026-10-05 with SymPy 1.14.0.