Home›Calculus 1›Basic antiderivatives›Problem 4.1280

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

Problem 4.1280 · medium Video

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

\[ \int \left(x + 1 + \frac{1}{x + 1}\right)\, dx \]
integralStart with the integral of the given function.✓ Proved
\[ = \int 1\, dx + \int x\, dx + \int \frac{1}{x + 1}\, dx \]
linearitySplit the integral into three separate parts.✓ Proved
\[ = \frac{x^{2}}{2} + x + \int \frac{1}{x + 1}\, dx \]
antiderivativeIntegrate the first two terms.✓ Proved
\[ = \frac{x^{2}}{2} + x + \ln{\left(x + 1 \right)} \]
antiderivativeIntegrate the final term.✓ Proved

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

Top: the graph of F = x²/2 + x + log(x + 1). A tangent line slides along it, and its slope is plotted below as it goes, tracing f = x + 1 + 1/(x + 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 + 1 = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x + 1 = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x + 1 = 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 (style) — [domain objection, downgraded to style] The antiderivative of 1/(x+1) is ln|x+1|, not ln(x+1); omitting the absolute value can mislead students about domain issues.
qwen3.6:27b-mlx: pass

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19
gpt-oss:20b: fail (style) 2026-10-19 — [domain objection, downgraded to style] The antiderivative of 1/(x+1) is ln|x+1|, not ln(x+1); omitting the absolute value can mislead students about domain issues.
qwen3.6:27b-mlx: pass 2026-10-18
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-18
gpt-oss:20b: pass 2026-10-18

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.