Integral of \( \displaystyle e^{3 x} \cos{\left(3 x \right)} \)
Problem 4.1215 · medium Video
Find \( \displaystyle \int e^{3 x} \cos{\left(3 x \right)} \, dx \). (Omit the constant of integration.)
- \[ \int e^{3 x} \cos{\left(3 x \right)}\, dx \]integralStart with the integral of the given function.✓ Proved
- \[ = \int \frac{\left(e^{3 i x} + e^{- 3 i x}\right) e^{3 x}}{2}\, dx \]rewriteUse the exponential form of the cosine function.≈ Checked numerically
- \[ = \int \left(\frac{e^{3 x - 3 i x}}{2} + \frac{e^{3 x + 3 i x}}{2}\right)\, dx \]algebraDistribute the exponential term.✓ Proved
- \[ = \int \frac{e^{3 x \left(1 - i\right)}}{2}\, dx + \int \frac{e^{3 x \left(1 + i\right)}}{2}\, dx \]linearitySplit the integral into two parts.✓ Proved
- \[ = \frac{\left(6 + 6 i\right) e^{3 x \left(1 - i\right)}}{72} + \frac{\left(6 - 6 i\right) e^{3 x \left(1 + i\right)}}{72} \]antiderivative simplifyIntegrate each exponential term. Simplify the denominators.✓ Proved
- \[ = \frac{\left(6 - 6 i\right) e^{3 x} e^{3 i x}}{72} + \frac{\left(6 + 6 i\right) e^{3 x} e^{- 3 i x}}{72} \]algebraSeparate the exponential terms.✓ Proved
- \[ = \frac{\left(1 + i\right) \left(6 - 6 i\right) \left(\left(1 - i\right) e^{3 i x} + \left(1 + i\right) e^{- 3 i x}\right) e^{3 x}}{144} \]algebraFind a common denominator.✓ Proved
- \[ = \frac{\left(e^{3 i x} - i e^{3 i x} + e^{- 3 i x} + i e^{- 3 i x}\right) e^{3 x}}{12} \]algebraExpand the numerator.✓ Proved
- \[ = \frac{\left(- i \left(e^{3 i x} - e^{- 3 i x}\right) + e^{3 i x} + e^{- 3 i x}\right) e^{3 x}}{12} \]algebraGroup terms to identify trigonometric functions.✓ Proved
- \[ = \frac{\left(2 \sin{\left(3 x \right)} + 2 \cos{\left(3 x \right)}\right) e^{3 x}}{12} \]rewrite algebraConvert the complex exponentials back to sine and cosine. Simplify the numerator using I**2 = -1.≈ Checked numerically
- \[ = \frac{\left(\sin{\left(3 x \right)} + \cos{\left(3 x \right)}\right) e^{3 x}}{6} \]simplifyFinal simplification.✓ Proved
Answer \( \frac{\sqrt{2} e^{3 x} \sin{\left(3 x + \frac{\pi}{4} \right)}}{6} + C \)
✓ Nihil obstat Lines: 12 proved, 2 checked numerically. 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 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left (1 - I)*(-exp(6*I*x) + (1 + I)*exp(3*I*x)*sin(3*x) + (1 + I)*exp(3*I*x)*cos(3*x) - I)*exp(3*x*(1 - I))/12; numeric agreement only, at 24 of 24 sampled points |
| 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 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 11 | ≈ Checked numerically | sympy 1.14.0 | sympy 1.14.0: simplify left (exp(6*I*x) - I*exp(6*I*x) - 2*sqrt(2)*exp(3*I*x)*sin(3*x + pi/4) + 1 + I)*exp(3*x*(1 - I))/12; numeric agreement only, at 24 of 24 sampled points |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 13 | ✓ 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 — The solution correctly applies complex exponentials to find the antiderivative. Each step adheres to the single-rule constraint, and the labels accurately describe the operations performed.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies complex exponentials to find the antiderivative. Each step adheres to the single-rule constraint, and the labels accurately describe the operations performed.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly applies complex exponentials to integrate the product of exponential and cosine functions. Each step adheres to the single-rule constraint, and the labels accurately describe the transformations performed.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies complex exponentials to find the antiderivative. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.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.