Home›Calculus 2›Partial fractions›Problem 4.1247

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

Problem 4.1247 · medium Video

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

\[ \int \frac{1}{\left(x - 1\right)^{2} - 4}\, dx \]
integralStart with the integral of the given function.✓ Proved
\[ = \int \frac{1}{x^{2} - 2 x - 3}\, dx \]
algebraExpand the squared term in the denominator.✓ Proved
\[ = \int \frac{1}{\left(x - 3\right) \left(x + 1\right)}\, dx \]
algebraFactor the quadratic expression in the denominator.✓ Proved
\[ = \int \left(- \frac{1}{4 \left(x + 1\right)} + \frac{1}{4 \left(x - 3\right)}\right)\, dx \]
partial-fractionsPerform partial fraction decomposition.✓ Proved
\[ = \frac{\int \frac{1}{x - 3}\, dx}{4} - \frac{\int \frac{1}{x + 1}\, dx}{4} \]
linearitySplit the integral into two parts using linearity.✓ Proved
\[ = \frac{\ln{\left(x - 3 \right)}}{4} - \frac{\ln{\left(x + 1 \right)}}{4} \]
antiderivativeIntegrate each term using the natural logarithm rule.✓ Proved

Answer \( \frac{\ln{\left(x - 3 \right)} - \ln{\left(x + 1 \right)}}{4} + C \)

Top: the graph of F = log(x - 3)/4 - log(x + 1)/4. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = 1/((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 undefined where x2 - 2*x - 3 = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x*2 - 2x - 3 = 0 undefined where x + 1 = 0 undefined where x - 3 = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x + 1 = 0 undefined where x - 3 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x + 1 = 0 undefined where x - 3 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x + 1 = 0 undefined where x - 3 = 0 log is undefined for non-positive arguments
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0 log is undefined for non-positive arguments
answer, a second way	✓ Proved	sympy 1.14.0	SymPy differentiated the stated antiderivative back to the integrand

Reviewers

gpt-oss:20b: fail (misleading) — The final antiderivative omits the absolute‑value bars that are required for the logarithm terms, which could mislead a student into thinking the result is valid for all real x.
qwen3.6:27b-mlx: pass — The solution correctly applies algebraic manipulation, partial fraction decomposition, linearity of integration, and standard antiderivative rules. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly applies algebraic manipulation, partial fraction decomposition, linearity of integration, and standard antiderivative rules. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
gpt-oss:20b: fail (misleading) 2026-10-19 — The final antiderivative omits the absolute‑value bars that are required for the logarithm terms, which could mislead a student into thinking the result is valid for all real x.
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies algebraic manipulation, partial fraction decomposition, and linearity of integration. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies algebraic expansion, factoring, partial fraction decomposition, linearity of integration, and standard antiderivatives. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
gpt-oss:20b: fail (style) 2026-10-16 — [domain objection, downgraded to style] The antiderivative of 1/(x‑3) and 1/(x+1) should be written with absolute values, i.e. ln|x‑3| and ln|x+1|. Omitting the absolute value signs can mislead a student into thinking the logarithm is defined for all real arguments.

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.