Home›Calculus 1›Substitution with a given u›Problem 4.935

Substitution with a given u

Problem 4.935 · hard

Evaluate \( \displaystyle \int 4 \sin^{6}{\left(x \right)} \cos{\left(x \right)}\, dx \) using the substitution \( \displaystyle u = \sin{\left(x \right)} \).

\[ \frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)} \]
So du = 1cos(1x) dx.✓ Proved
In terms of u the integral becomes ∫ 4*u**6 du.
Reviewed
\[ \frac{d}{d u} \frac{4 u^{7}}{7} = 4 u^{6} \]
An antiderivative in u is 4*u**7/7; differentiating confirms it.✓ Proved
\[ \frac{4 \sin^{7}{\left(x \right)}}{7} \]
Replace u by sin(x).✓ Proved
\[ \frac{d}{d x} \frac{4 \sin^{7}{\left(x \right)}}{7} = 4 \sin^{6}{\left(x \right)} \cos{\left(x \right)} \]
Differentiating the answer returns the integrand.✓ Proved

Answer \( \frac{4 \sin^{7}{\left(x \right)}}{7} + C \)

✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
2	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
4	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
5	✓ 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	a difference quotient of the answer matches the integrand at x = 0.4, 0.9, 1.3

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the substitution method, handles the differential transformation accurately, and verifies the result.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-12
qwen3.6:27b-mlx: pass 2026-10-12 — The solution correctly applies the substitution method, handles the differential transformation accurately, and verifies the result.
gpt-oss:20b: pass 2026-10-12
qwen3.6:27b-mlx: pass 2026-10-12

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/indefinite_substitution, checked 2026-10-12 with SymPy 1.14.0.