Home›Calculus 2›Partial fractions›Problem 4.1476

Integral of \( \displaystyle \frac{x + 6}{x + \left(x + 1\right)^{2} - 1} \)

Problem 4.1476 · medium Video

Find \( \displaystyle \int \frac{x + 6}{x + \left(x + 1\right)^{2} - 1} \, dx \). (Omit the constant of integration.)

\[ \int \frac{x + 6}{x + \left(x + 1\right)^{2} - 1}\, dx \]
simplifyStart with the given integral.✓ Proved
\[ = \int \frac{x + 6}{x^{2} + 3 x}\, dx \]
algebra algebraExpand the squared term. Combine like terms in the denominator.✓ Proved
\[ = \int \frac{x + 6}{x \left(x + 3\right)}\, dx \]
algebraFactor the denominator.✓ Proved
\[ = \int \left(\frac{1}{x + 3} + \frac{6}{x \left(x + 3\right)}\right)\, dx \]
linearity simplifySplit the integrand into two parts. Simplify the second term by canceling x.✓ Proved
\[ = \int \frac{6}{x \left(x + 3\right)}\, dx + \int \frac{1}{x + 3}\, dx \]
linearitySplit the integral again.✓ Proved
\[ = \int \frac{1}{x + 3}\, dx + \int \left(- \frac{2}{x + 3} + \frac{2}{x}\right)\, dx \]
partial-fractionsApply partial fraction decomposition to the first term.✓ Proved
\[ = \int \frac{2}{x}\, dx + \int \frac{1}{x + 3}\, dx - \int \frac{2}{x + 3}\, dx \]
linearityDistribute the integral sign.✓ Proved
\[ = \int \frac{2}{x}\, dx - \int \frac{1}{x + 3}\, dx \]
algebraCombine the integrals of the (x+3) terms.✓ Proved
\[ = 2 \ln{\left(x \right)} - \ln{\left(x + 3 \right)} \]
antiderivativeEvaluate the remaining integrals.✓ Proved

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

Top: the graph of F = 2log(x) - log(x + 3). A tangent line slides along it, and its slope is plotted below as it goes, tracing f = (x + 6)/(x + (x + 1)² - 1).

✓ 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 + (x + 1)2 - 1 = 0 undefined where x2 + 3*x = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x*2 + 3x = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x*2 + 3x = 0 undefined where x + 3 = 0 undefined where x = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x + 3 = 0 undefined where x = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x + 3 = 0 undefined where x = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x + 3 = 0 undefined where x = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x + 3 = 0 undefined where x = 0
9	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x + 3 = 0 undefined where x = 0
10	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x + 3 = 0 undefined where x = 0
11	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x + 3 = 0 undefined where x = 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 simplifies the integrand, applies partial fraction decomposition, and integrates term by term. Each step adheres to the single-rule constraint and uses valid labels from the fixed vocabulary.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-11-09 — The solution correctly simplifies the integrand, applies partial fraction decomposition, and integrates term by term. Each step adheres to the single-rule constraint and uses valid labels from the fixed vocabulary.
gpt-oss:20b: pass 2026-11-09
qwen3.6:27b-mlx: pass 2026-11-08 — The solution correctly simplifies the integrand, applies partial fraction decomposition, and evaluates the integrals. Each step adheres to the single-rule constraint and uses valid labels from the fixed vocabulary.
gpt-oss:20b: pass 2026-11-08
qwen3.6:27b-mlx: pass 2026-11-08 — The solution correctly simplifies the integrand, applies partial fraction decomposition, and integrates term-by-term. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
gpt-oss:20b: pass 2026-11-08

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 2026-11-09 with SymPy 1.14.0.