Answer:
14. ½ atan(x/5) − 5x / (50 + 2x²)
15. 12 asin(x/6) + x/3 √(36 − x²)
Explanation:
14. x = 5 tan θ
θ/2 − sin(2θ)/4
Use double angle formula:
sin(2θ) = 2 tan θ / (1 + tan² θ)
tan θ = x / 5, so:
sin(2θ) = (2x/5) / (1 + x²/25)
sin(2θ) = 10x / (25 + x²)
Substitute:
θ/2 − (10x / (25 + x²)) / 4
θ/2 − 5x / (50 + 2x²)
½ atan(x/5) − 5x / (50 + 2x²)
15. x = 6 sin θ
12θ + 6 sin 2θ
Use double angle formula:
sin(2θ) = 2 sin θ cos θ
sin(2θ) = 2 sin θ √(1 − sin² θ)
sin θ = x / 6, so:
sin(2θ) = (x/3) √(1 − x²/36)
sin(2θ) = x/18 √(36 − x²)
Substitute:
12θ + 6 (x/18 √(36 − x²))
12θ + x/3 √(36 − x²)
12 asin(x/6) + x/3 √(36 − x²)