Integral of \( \displaystyle e^{3 x - 3} \)
Problem 4.1278 · easy Video
Find \( \displaystyle \int e^{3 x - 3} \, dx \). (Omit the constant of integration.)
- \[ \int e^{3 x - 3}\, dx \]integralStart with the integral of the function.✓ Proved
- \[ = \int \frac{e^{3 x}}{e^{3}}\, dx \]algebraUse the property exp(a + b) = exp(a) * exp(b).✓ Proved
- \[ = \frac{\int e^{3 x}\, dx}{e^{3}} \]linearityPull the constant factor out of the integral.✓ Proved
- \[ = \frac{e^{3 x}}{3 e^{3}} \]antiderivativeIntegrate the exponential function.✓ Proved
- \[ = \frac{e^{3 x - 3}}{3} \]simplifyCombine the terms back into a single exponent.✓ Proved
Answer \( \frac{e^{3 x - 3}}{3} + C \)
✓ 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 |
| 5 | ✓ 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: passqwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-19gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-18gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-18gpt-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.