Home›Calculus 1›Basic antiderivatives›Problem 4.186

Integral of \( \displaystyle \sqrt{x - 1} \)

Problem 4.186 · easy Video

Find \( \displaystyle \int \sqrt{x - 1} \, dx \). (Omit the constant of integration.)

\[ \int \sqrt{x - 1}\, dx \]
integral rewriteStart with the integral of the given function. Rewrite the square root as a fractional exponent.✓ Proved
\[ = \frac{2 \left(x - 1\right)^{\frac{3}{2}}}{3} \]
antiderivative simplifyApply the power rule for integration. Simplify the coefficient.✓ Proved

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

Top: the graph of F = 2(x - 1)^(3/2)/3. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = sqrt(x - 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
4	✓ 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: pass — The solution correctly applies the power rule for integration via a simple substitution (implicitly handled by the direct application of the antiderivative rule to the composite function, which is standard in this context or covered by the 'substitution' rule if explicitly stated, but here 'antiderivative' is acceptable as it names the core operation). Each step changes only one aspect of the expression and uses valid labels from the vocabulary.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies the power rule for integration via a simple substitution (implicitly handled by the direct application of the antiderivative rule to the composite function, which is standard in this context or covered by the 'substitution' rule if explicitly stated, but here 'antiderivative' is acceptable as it names the core operation). Each step changes only one aspect of the expression and uses valid labels from the vocabulary.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: fail (error) 2026-09-29 — Step 3 applies the power rule for integration directly to the composite function (x-1)^(1/2) without accounting for the chain rule factor (or performing a substitution). The antiderivative of (x-1)^(1/2) is not (x-1)^(3/2)/(3/2); the step skips the necessary substitution u = x - 1 or fails to justify why the inner derivative (which is 1) allows this shortcut, making the application of the 'antiderivative' rule formally incorrect for a composite function.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: pass 2026-09-29
gpt-oss:20b: pass 2026-09-29

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 2026-10-03 with SymPy 1.14.0.