Home›Calculus 1›u-substitution›Problem 4.1120

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

Problem 4.1120 · medium Video

Find \( \displaystyle \int \frac{x - 1}{\sqrt{\left(x - 1\right)^{2} + 4}} \, dx \). (Omit the constant of integration.)

\[ \int \frac{x - 1}{\sqrt{\left(x - 1\right)^{2} + 4}}\, dx \]
integral rewriteStart with the integral of the given function. Rewrite the square root using a fractional exponent.✓ Proved
\[ = \int \frac{2 x - 2}{2 \sqrt{\left(x - 1\right)^{2} + 4}}\, dx \]
algebraRewrite the numerator to match the derivative of the inner function.✓ Proved
\[ = \sqrt{\left(x - 1\right)^{2} + 4} \]
antiderivativeThe integral follows the power rule for the chain rule.✓ Proved

Answer \( \sqrt{\left(x - 1\right)^{2} + 4} + C \)

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

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 undefined where (x - 1)**2 + 4 = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where (x - 1)**2 + 4 = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where (x - 1)**2 + 4 = 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 4 applies the antiderivative rule but fails to account for the constant factor 1/2 introduced in Step 3, resulting in an answer that is twice the correct value. Additionally, Step 3 applies algebraic manipulation to the integrand but does not pull the constant factor out via linearity, violating the one-rule-per-step constraint.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — Step 4 applies the antiderivative rule but fails to account for the constant factor 1/2 introduced in Step 3, resulting in an answer that is twice the correct value. Additionally, Step 3 applies algebraic manipulation to the integrand but does not pull the constant factor out via linearity, violating the one-rule-per-step constraint.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly applies substitution via algebraic manipulation in step 3 and identifies the antiderivative in step 4. Each step changes only one aspect of the expression, and the labels are appropriate.
gpt-oss:20b: pass 2026-10-14

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 2026-10-17 with SymPy 1.14.0.