∫ -∞ e -x² dx

∫ -∞ e -x² dx "Math is easy" 🤓 I = ∫ -∞ e -x² dx "Math is easy" 🤓 I² = (∫ -∞ e -x² dx)² "Math is easy" 🤓 I² = (∫ -∞ e -x² dx)(∫ -∞ e -y² dy) "Math is easy" 🤓 I² = ∫∫ -∞ -∞ e -x² e -y² dx dy "Math is easy" 🤓 I² = ∫∫ -∞ -∞ e -(x²+y²) dx dy "Math is easy" 🤓 ∫∫ -∞ -∞ e -((r cos θ)²+(r sin θ)²) dx dy "Math is easy" 🤓 ∫∫ -∞ -∞ ∂ ∂r x(r, θ) ∂ ∂θ x(r, 0) ∂ ∂r y(r, 0) ∂ ∂θ y(r, 0) e -((r cos θ)²+(r sin θ)²) d(r, θ) "Math is easy" 🤓 ∫∫ -∞ -∞ ∂ ∂r r cos θ ∂ ∂θ r cos θ ∂ ∂r r sin θ ∂ ∂θ r sin θ e -((r cos θ)²+(r sin θ)²) d(r, θ) "Math is easy" 🤓 ∫∫ cos θ sin θ -r sin θ r cos θ e -((r cos θ)²+(r sin θ)²) d(r, θ) "Math is easy" 🤓 ∫∫ cos θ -r sin θ sin θ r cos θ e -r²(cos² θ+sin² θ) d(r, θ) "Math is easy" 🤓 ∫∫ -∞ ∞ ∫ 0 2π |cos θ -r sin θ sin θ r cos θ| e -r² dθ dr "Math is easy" 🤓 ∫ 0 ∞ ∫ 0 2π [r cos² θ - (-r sin² θ)]e -r² dθ dr "Math is easy" 🤓 ∫ 0 ∞ ∫ 0 2π [r cos² θ + r sin² θ]e -r² dθ dr "Math is easy" 🤓 ∫ 0 ∞ ∫ 0 2π r(cos² θ + sin² θ)e -r² dθ dr "Math is easy" 🤓 ∫ 0 ∞ ∫ 0 2π r e -r² dθ dr "Math is easy" 🤓 (∫ 0 ∞ r e -r² dr)(∫ 0 2π dθ) "Math is easy" 🤓 (1/2 ∫ -∞ 0 e -u² du)(|θ| 0 2π) "Math is easy" 🤓 I² = 1/2 ((-1/2 e -∞² - (-1/2 e 0²)) (2π) "Math is easy" 🤓 I² = 1/2 (-1/2 (0 - 1)) (2π) "Math is easy" 🤓 I² = 1/2 (1/2) (2π) "Math is easy" 🤓 I² = π "Math is easy" 🤓 I = √π "Math is easy" 🤓