Home›Calculus 1›Basic antiderivatives›Problem 4.1218

Integral of \( \displaystyle e^{6 x - 3} \)

Problem 4.1218 · easy Video

Find \( \displaystyle \int e^{6 x - 3} \, dx \). (Omit the constant of integration.)

\[ \int e^{6 x - 3}\, dx \]
integralStart with the integral of the given function.✓ Proved
\[ = \int \frac{e^{6 x}}{e^{3}}\, dx \]
algebraUse the property of exponents to split the term.✓ Proved
\[ = \frac{\int e^{6 x}\, dx}{e^{3}} \]
linearityFactor out the constant term.✓ Proved
\[ = \frac{e^{6 x}}{6 e^{3}} \]
antiderivativeApply the rule for integrating an exponential function.✓ Proved

Answer \( \frac{e^{6 x - 3}}{6} + C \)

Top: the graph of F = e^(6x - 3)/6. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = e^(6x - 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
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0
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, linearity, and the standard antiderivative rule for exponential functions. 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 2026-10-18 — The solution correctly applies algebraic manipulation, linearity, and the standard antiderivative rule for exponential functions. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies algebraic manipulation to separate the constant factor, uses linearity to pull it out of the integral, and applies the standard antiderivative rule for exponential functions. Each step changes only one aspect of the expression and uses valid labels.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies algebraic manipulation, linearity, and the standard antiderivative rule for exponential functions in distinct steps.
gpt-oss:20b: pass 2026-10-15

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 2026-10-18 with SymPy 1.14.0.