Problem Analysis #calculus_24

Motif 3 · Run 1

Python ● execution succeeded

Correct

6.74s

1,976 toks

Infron · Motif 3 · 46문항 × 3회 · Python TAR. 131,072 output tokens · 기본 1,800초 · 6+1 turns. ACC는 전체 138회, Pass@1은 각 문항의 첫 유효 시도, Pass@3는 3회 내 정답으로 계산합니다.

Problem Statement

$\int_0^{\pi/2}\sqrt{\sin x-\sin^3x}\,dx$ 의 값은?

① $\frac16$ ② $\frac13$ ③ $\frac12$ ④ $\frac23$ ⑤ $\frac56$

최종 답은 선택지 번호 1~5 중 하나를 $\boxed{}$ 안에 쓰세요.

Ground Truth

Recorded model response / solution

Reasoning Process

Recorded Python call:

import mpmath as mp
f = lambda x: mp.sqrt(mp.sin(x) - mp.sin(x)**3)
print(mp.quad(f, [0, mp.pi/2]))

Tool Output

Execution succeeded 0.666666666666667

Reasoning Process

\int_0^{\pi/2}\sqrt{\sin x-\sin^3x}\,dx = \int_0^{\pi/2}\sqrt{\sin x\cos^2x}\,dx = \int_0^{\pi/2}\cos x\sqrt{\sin x}\,dx.

Let $u=\sin x$ , so $du=\cos x\,dx$ . Then

\int_0^1 \sqrt{u}\,du = \left[\frac{2}{3}u^{3/2}\right]_0^1 = \frac{2}{3}.

따라서 정답은 ④입니다.

\boxed{4}

Final Model Output

Comparative Analysis (Pcalculus_24) (Average across runs)

Averages use reported measurements only. Missing usage is not treated as zero.

Avg Time Taken

Solar Pro 4

35.5

MiMo V2.6 Pro

32.1

Union Alpha

28.6

Grok 4.7

16.2

Motif 3

14.9

Muse Spark 1.3

10.3

DeepSeek V4.1 Flash

4.9

Solar Mini 4 Preview

3.8

Avg Token Usage

Grok 4.7

7,674

Solar Pro 4

7,009

Motif 3

5,636

MiMo V2.6 Pro

4,422

Muse Spark 1.3

4,236

Union Alpha

4,000

Solar Mini 4 Preview

2,598

DeepSeek V4.1 Flash

2,508