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Home » Let $ y = \cos x $, where $ y \in [-1, 1] $. The function becomes: - Get link 4share

Let $ y = \cos x $, where $ y \in [-1, 1] $. The function becomes: - Get link 4share

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 \int_0^1 \int_0^{\min(0.2, 1 - u)} 30 s^4 \, ds \, du

📌 Split into two regions: $ 0 \leq u \leq 0.8 $, where $ 1 - u \geq 0.2 $, so $ s $ goes to 0.2; for $ u > 0.8 $, $ 1 - u < 0.2 $, so $ s $ goes to $ 1 - u $

📌 = \int_0^{0.8} \int_0^{0.2} 30 s^4 ds \, du + \int_{0.8}^1 \int_0^{1 - u} 30 s^4 ds \, du

📌 First integral:

📌 \int_0^{0.2} 30 \cdot \frac{s^5}{5} ds = 6 \int_0^{0.2} s^5 ds = 6 \cdot \frac{s^6}{6} \Big|_0^{0.2} = (0.2)^6 = \left(\frac{1}{5}\right)^6 = \frac{1}{15625}

📌 Multiply by 1 (no outer u): $ \frac{1}{15625} $

📌 Second integral:

📌 \int_{0.8}^1 30 \cdot \frac{(1 - u)^5}{5} du = 6 \int_{0.8}^1 (1 - u)^5 du

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