Home›Calculus 2›Integration by parts›Problem 4.1404

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

Problem 4.1404 · easy Video

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

\[ \int e^{x} \sin{\left(x \right)}\, dx \]
integralStart with the integral of the integrand.✓ Proved
\[ = - e^{x} \cos{\left(x \right)} + \int e^{x} \cos{\left(x \right)}\, dx \]
partsApply integration by parts with u = exp(x) and dv = sin(x) dx.✓ Proved
\[ = e^{x} \sin{\left(x \right)} - e^{x} \cos{\left(x \right)} - \int e^{x} \sin{\left(x \right)}\, dx \]
partsApply integration by parts to the remaining integral with u = exp(x) and dv = cos(x) dx.✓ Proved
\[ = 0.5 e^{x} \sin{\left(x \right)} - 0.5 e^{x} \cos{\left(x \right)} \]
algebraSolve the equation for the integral.✓ Proved

Answer \( \frac{e^{x} \sin{\left(x \right)} - e^{x} \cos{\left(x \right)}}{2} + C \)

Top: the graph of F = e^(x)·sin(x)/2 - e^(x)·cos(x)/2. A tangent line slides along it, and its slope is plotted below as it goes, tracing f = e^(x)·sin(x).

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. The reviewers disagree about how one step is explained; every verdict is in the receipt. 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 integration by parts twice and then uses algebra to solve for the original integral. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-11-01 — The solution correctly applies integration by parts twice and then uses algebra to solve for the original integral. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.
gpt-oss:20b: pass 2026-11-01
qwen3.6:27b-mlx: pass 2026-10-31
gpt-oss:20b: pass 2026-10-31
qwen3.6:27b-mlx: pass 2026-10-31 — The solution correctly applies integration by parts twice and then uses algebra to solve for the original integral. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
gpt-oss:20b: fail (error) 2026-10-31 — Step 2 incorrectly applies integration by parts: the correct result is e^x sin(x) - ∫e^x cos(x)dx, not -e^x cos(x)+∫e^x cos(x)dx. The integrand term e^x sin(x) is missing and the sign of the remaining integral is wrong.

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 2026-11-01 with SymPy 1.14.0.