Home›Calculus 2›Partial fractions›Problem 4.1285

Integral of \( \displaystyle \frac{x + 4}{x + \left(x - 1\right)^{2} - 3} \)

Problem 4.1285 · medium Video

Find \( \displaystyle \int \frac{x + 4}{x + \left(x - 1\right)^{2} - 3} \, dx \). (Omit the constant of integration.)

\[ \int \frac{x + 4}{x + \left(x - 1\right)^{2} - 3}\, dx \]
integralStart with the integral of the given function.✓ Proved
\[ = \int \frac{x + 4}{x^{2} - x - 2}\, dx \]
algebra simplifyExpand the squared term. Combine like terms in the denominator.✓ Proved
\[ = \int \frac{x + 4}{\left(x - 2\right) \left(x + 1\right)}\, dx \]
algebraFactor the quadratic denominator.✓ Proved
\[ = \int \left(- \frac{1}{x + 1} + \frac{2}{x - 2}\right)\, dx \]
partial-fractionsApply partial fraction decomposition.✓ Proved
\[ = 2 \int \frac{1}{x - 2}\, dx - \int \frac{1}{x + 1}\, dx \]
linearitySplit the integral into two parts.✓ Proved
\[ = 2 \ln{\left(x - 2 \right)} - \ln{\left(x + 1 \right)} \]
antiderivativeIntegrate each term.✓ Proved

Answer \( 2 \ln{\left(x - 2 \right)} - \ln{\left(x + 1 \right)} + C \)

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

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 + (x - 1)2 - 3 = 0 undefined where x2 - x - 2 = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x**2 - x - 2 = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x**2 - x - 2 = 0 undefined where x + 1 = 0 undefined where x - 2 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x + 1 = 0 undefined where x - 2 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x + 1 = 0 undefined where x - 2 = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x + 1 = 0 undefined where x - 2 = 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: pass
qwen3.6:27b-mlx: pass

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-20
gpt-oss:20b: pass 2026-10-20
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly applies algebraic simplification, 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 (misleading) 2026-10-19 — The antiderivative step omits the absolute value in the logarithms; a student would learn that ∫1/(x‑2)dx = log(x‑2) instead of log|x‑2|, which is incorrect for all real x.
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly applies algebraic simplification, 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: pass 2026-10-19

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 2026-10-20 with SymPy 1.14.0.