Home›Calculus 1›u-substitution›Problem 4.1334

Integral of \( \displaystyle e^{3 x} \sin{\left(e^{3 x} \right)} \)

Problem 4.1334 · medium Video

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

\[ \int e^{3 x} \sin{\left(e^{3 x} \right)}\, dx \]
integral algebraStart with the integral of the function. Multiply and divide by 3 to create the derivative of the inner function.✓ Proved
\[ = \frac{\int 3 e^{3 x} \sin{\left(e^{3 x} \right)}\, dx}{3} \]
linearity rewritePull the constant factor out of the integral. Rearrange terms to highlight the substitution pattern.✓ Proved
\[ = - \frac{\cos{\left(e^{3 x} \right)}}{3} \]
antiderivativeIntegrate the expression.✓ Proved

Answer \( - \frac{\cos{\left(e^{3 x} \right)}}{3} + C \)

Top: the graph of F = -cos(e^(3x))/3. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = e^(3x)·sin(e^(3x)).

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies substitution via algebraic manipulation and linearity, followed by recognizing the antiderivative of the composite function. Each step changes only one aspect of the expression and uses valid labels.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-24 — The solution correctly applies substitution via algebraic manipulation and linearity, followed by recognizing the antiderivative of the composite function. Each step changes only one aspect of the expression and uses valid labels.
gpt-oss:20b: pass 2026-10-24
qwen3.6:27b-mlx: pass 2026-10-23 — The solution correctly applies substitution via algebraic manipulation and linearity, followed by direct integration of the composite function. All steps adhere to the single-rule constraint and use valid labels.
gpt-oss:20b: pass 2026-10-23
qwen3.6:27b-mlx: pass 2026-10-23
gpt-oss:20b: pass 2026-10-23

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 2026-10-24 with SymPy 1.14.0.