Home›Calculus 1›u-substitution›Problem 4.1249

Integral of \( \displaystyle \left(4 x + 2\right) \sqrt{\left(2 x + 1\right)^{2} + 1} \)

Problem 4.1249 · hard Video

Find \( \displaystyle \int 2 \left(2 x + 1\right) \sqrt{\left(2 x + 1\right)^{2} + 1} \, dx \). (Omit the constant of integration.)

\[ \int \left(4 x + 2\right) \sqrt{\left(2 x + 1\right)^{2} + 1}\, dx \]
integralStart with the integral of the given function.✓ Proved
\[ = \int \left(4 x + 2\right) \sqrt{4 x^{2} + 4 x + 2}\, dx \]
algebraExpand the expression inside the square root.✓ Proved
\[ = \int \left(4 x + 2\right) \sqrt{\left(2 x + 1\right)^{2} + 1}\, dx \]
rewrite algebraRewrite the radicand in terms of (2x+1)^2 + 1. Distribute the 2 into the linear term.✓ Proved
\[ = \frac{\left(\left(2 x + 1\right)^{2} + 1\right)^{\frac{3}{2}}}{3} \]
antiderivativeIntegrate using the power rule for the substitution u = (2x+1)^2 + 1.✓ Proved

Answer \( \frac{\left(\left(2 x + 1\right)^{2} + 1\right)^{\frac{3}{2}}}{3} + C \)

Top: the graph of F = ((2x + 1)² + 1)^(3/2)/3. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (4x + 2)·sqrt((2x + 1)² + 1).

✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation. How we verify

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
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0
answer, a second way	✓ Proved	sympy 1.14.0	SymPy differentiated the stated antiderivative back to the integrand

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (style) — Steps 2 and 3 are redundant and violate the one-change-per-step principle by expanding and then immediately re-contracting the expression without advancing the solution. Step 5 applies both substitution and integration in a single step, violating the granularity constraint.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-10-19 — Steps 2 and 3 are redundant and violate the one-change-per-step principle by expanding and then immediately re-contracting the expression without advancing the solution. Step 5 applies both substitution and integration in a single step, violating the granularity constraint.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (style) 2026-10-16 — Steps 2 and 3 are redundant and violate the one-change-per-step principle by expanding and then immediately re-contracting the radicand without progress. Step 5 applies substitution and integration in a single step, violating the granularity constraint.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-16
gpt-oss:20b: pass 2026-10-16

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 2026-10-19 with SymPy 1.14.0.