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A tricky radical integral made simple 🔥📐 At first glance, ∫ √((1 − √x)/(1 √x)) dx looks scary 😵‍💫 But with a smart trig sub: √x = cos(2θ) dx = -4 cos(2θ) sin(2θ) dθ The integrand simplifies to tan(θ), leading to: -8 ∫ sin²(θ) cos(2θ) dθ Using identities: cos(2θ) = 1 - 2 sin²(θ) It becomes -8 ∫ (sin²(θ) - 2 sin⁴(θ)) dθ Integrate term-by-term with power-reduction: Result: 2θ - 2 sin(2θ) (1/2) sin(4θ) C Back-sub: θ = (1/2) arccos(√x) sin(2θ) = √(1 - x) sin(4θ) = 2 √x √(1 - x) Final closed form: arccos(√x) - 2√(1 - x) √(x(1 - x)) C 💡 Master these tricks to conquer tough integrals! 🚀 Follow @engineer_know for more calculus breakdowns 📘 #Calculus #Integration #TrigonometricSubstitution #MathReels #EngineeringMath #LearnMath #MathSimplified

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