Home›Calculus 1›u-substitution›Problem 4.1429

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

Problem 4.1429 · medium Video

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

\[ \int \left(2 x + 2\right) \sqrt{\left(x + 1\right)^{2} + 1}\, dx \]
integralStart with the integral of the given function.✓ Proved
\[ = \int \left(2 x + 2\right) \sqrt{x^{2} + 2 x + 2}\, dx \]
algebraExpand the square inside the square root.✓ Proved
\[ = \frac{2 \left(\left(x + 1\right)^{2} + 1\right)^{\frac{3}{2}}}{3} \]
antiderivativeUse the power rule for integration via substitution u = (x+1)**2 + 1.✓ Proved

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

Top: the graph of F = 2(x² + 2x + 2)^(3/2)/3. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (2x + 2)·sqrt((x + 1)² + 1).

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. 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
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 (error) — Step 3 applies both the substitution method and the power rule for integration in a single step, violating the one-rule-per-step constraint. Additionally, the label 'antiderivative' is not in the allowed vocabulary; 'substitution' or 'algebra' would be appropriate depending on how the step is split.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-11-04 — Step 3 applies both the substitution method and the power rule for integration in a single step, violating the one-rule-per-step constraint. Additionally, the label 'antiderivative' is not in the allowed vocabulary; 'substitution' or 'algebra' would be appropriate depending on how the step is split.
gpt-oss:20b: pass 2026-11-04
qwen3.6:27b-mlx: fail (style) 2026-11-03 — Step 3 applies substitution and the power rule for integration simultaneously, violating the one-rule-per-step constraint. It should be split into a substitution step and an integration step.
gpt-oss:20b: pass 2026-11-03
qwen3.6:27b-mlx: fail (error) 2026-11-03 — Step 3 jumps directly to the final answer without showing the substitution step, violating the 'one change per step' rule. Additionally, the label 'antiderivative' is not in the allowed vocabulary; 'substitution' or 'algebra' would be appropriate.
gpt-oss:20b: pass 2026-11-03

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 2026-11-04 with SymPy 1.14.0.