Home›Calculus 2›Partial fractions›Problem 4.1402

Integral of \( \displaystyle \frac{x}{\left(x + 1\right) \left(x + 3\right)} \)

Problem 4.1402 · medium Video

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

\[ \int \frac{x}{\left(x + 1\right) \left(x + 3\right)}\, dx \]
integral algebraStart with the integral of the given function. Rewrite the numerator to facilitate decomposition.✓ Proved
\[ = \int \left(\frac{1}{x + 3} - \frac{1}{\left(x + 1\right) \left(x + 3\right)}\right)\, dx \]
linearity simplifySplit the integral into two parts. Simplify the first term by canceling (x + 1).✓ Proved
\[ = \int \left(\frac{1}{2 x + 6} - \frac{1}{2 x + 2} + \frac{1}{x + 3}\right)\, dx \]
partial-fractions algebraApply partial fraction decomposition to the second term. Distribute the negative sign.✓ Proved
\[ = \int \left(\frac{3}{2 x + 6} - \frac{1}{2 x + 2}\right)\, dx \]
algebraCombine the terms involving (x + 3).✓ Proved
\[ = - \frac{\ln{\left(x + 1 \right)}}{2} + \frac{3 \ln{\left(x + 3 \right)}}{2} \]
antiderivativeThe antiderivative of 1/(x+a) is log(x+a).✓ Proved

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

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

✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation. 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 = 0 undefined where x + 3 = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 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 undefined where 2x + 6 = 0 undefined where 2x + 2 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 2x + 6 = 0 undefined where 2x + 2 = 0 undefined where x + 3 = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 2x + 6 = 0 undefined where 2x + 2 = 0 undefined where x + 3 = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 2x + 6 = 0 undefined where 2x + 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 — The solution correctly applies algebraic manipulation, partial fraction decomposition, 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 2026-11-01 — The solution correctly applies algebraic manipulation, partial fraction decomposition, and standard antiderivative rules. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
gpt-oss:20b: pass 2026-11-01
qwen3.6:27b-mlx: pass 2026-10-31 — The solution correctly applies partial fraction decomposition and linearity of integration. Each step changes only one aspect of the expression, and the labels accurately reflect the operations performed.
gpt-oss:20b: pass 2026-10-31
qwen3.6:27b-mlx: pass 2026-10-31 — The solution correctly applies partial fraction decomposition and linearity of integration. Each step changes only one aspect of the expression and uses valid labels from the allowed vocabulary.
gpt-oss:20b: pass 2026-10-31

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 2026-11-01 with SymPy 1.14.0.