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∫(0,1) x arcsinx dx
= ∫(0,1) arcsinx d(x²/2)
= (1/2)x² arcsinx |(0,1) - (1/2)∫(0,1) x²/√(1 - x²) dx
= (1/2)(1)(π/2) + (1/2)∫(0,1) [(1 - x²) - 1]/√(1 - x²) dx
= π/4 + (1/2)∫(0,1) √(1 - x²) dx - (1/2)∫(0,1) dx/√(1 - x²) dx
= π/4 + (1/2)(1/4)π(1)² - (1/2)arcsinx |(0,1)
= π/4 + π/8 - (1/2)(π/2)
= π/8
= ∫(0,1) arcsinx d(x²/2)
= (1/2)x² arcsinx |(0,1) - (1/2)∫(0,1) x²/√(1 - x²) dx
= (1/2)(1)(π/2) + (1/2)∫(0,1) [(1 - x²) - 1]/√(1 - x²) dx
= π/4 + (1/2)∫(0,1) √(1 - x²) dx - (1/2)∫(0,1) dx/√(1 - x²) dx
= π/4 + (1/2)(1/4)π(1)² - (1/2)arcsinx |(0,1)
= π/4 + π/8 - (1/2)(π/2)
= π/8
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