Answer:
π³/1296
Explanation:
∫ [ (asin x)² dx / √(4 − 4x²) ]
Factoring the denominator:
∫ [ (asin x)² dx / √(4 (1 − x²)) ]
∫ [ (asin x)² dx / (2√(1 − x²)) ]
½ ∫ [ (asin x)² dx / √(1 − x²) ]
If u = asin x, then du = dx / √(1 − x²).
½ ∫ u² du
⅙ u³ + C
Substitute back:
⅙ (asin x)³ + C
Evaluate between x=0 and x=½.
⅙ (asin ½)³ − ⅙ (asin 0)³
⅙ (π/6)³ − ⅙ (0)³
π³/1296