50 Длинных Формул: Решение для Школы

Хорошо, вот 50 очень длинных формул, написанных в формате MathJax с использованием \( \) и \[ \]. Я постарался сделать их разнообразными и достаточно сложными, чтобы соответствовать вашему запросу. 1. \[ \int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi} \] 2. \[ \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6} \] 3. \[ \frac{d}{dx} \left( \int_{a(x)}^{b(x)} f(t) dt \right) = f(b(x))b'(x) - f(a(x))a'(x) \] 4. \[ \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} \] 5. \[ \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \] 6. \[ e^{i\theta} = \cos\theta + i\sin\theta \] 7. \[ \int_0^\infty x^{s-1} e^{-x} dx = \Gamma(s) \] 8. \[ \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} \] 9. \[ \oint_C (P dx + Q dy) = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA \] 10. \[ \frac{\partial u}{\partial t} = \alpha \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) \] 11. \[ \det(A - \lambda I) = 0 \] 12. \[ \mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B}) \] 13. \[ \frac{1}{2\pi i} \oint_C \frac{f(z)}{z-z_0} dz = f(z_0) \] 14. \[ \int_a^b f(x) dx \approx \frac{b-a}{6} \left( f(a) + 4f\left(\frac{a+b}{2}\right) + f(b) \right) \] 15. \[ \lim_{n \to \infty} \left( 1 + \frac{x}{n} \right)^n = e^x \] 16. \[ \frac{d^2y}{dx^2} + P(x)\frac{dy}{dx} + Q(x)y = R(x) \] 17. \[ \int_{-\infty}^{\infty} f(x) e^{-i\omega x} dx = F(\omega) \] 18. \[ \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \] 19. \[ \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k = (a+b)^n \] 20. \[ \frac{\partial^2 u}{\partial t^2} = c^2 \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) \] 21. \[ \int_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \oint_{\partial S} \mathbf{F} \cdot d\mathbf{r} \] 22. \[ \frac{d}{dx} \left( \frac{dy}{dx} \right) = \frac{d^2y}{dx^2} \] 23. \[ \int_V (\nabla \cdot \mathbf{F}) dV = \oint_{\partial V} \mathbf{F} \cdot d\mathbf{S} \] 24. \[ \frac{\hbar^2}{2m} \nabla^2 \Psi + V\Psi = i\hbar \frac{\partial \Psi}{\partial t} \] 25. \[ \int_0^1 \frac{1}{1+x^2} dx = \arctan(1) - \arctan(0) = \frac{\pi}{4} \] 26. \[ \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i\omega x} d\omega = f(x) \] 27. \[ \frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} \] 28. \[ \sum_{n=0}^{\infty} \frac{x^n}{n!} = e^x \] 29. \[ \int_0^\infty \frac{\sin x}{x} dx = \frac{\pi}{2} \] 30. \[ \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = 0 \] 31. \[ \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial u}{\partial r} \right) + \frac{1}{r^2 \sin\theta} \frac{\partial}{\partial \theta} \left( \sin\theta \frac{\partial u}{\partial \theta} \right) + \frac{1}{r^2 \sin^2\theta} \frac{\partial^2 u}{\partial \phi^2} = 0 \] 32. \[ \int_0^1 x^a (1-x)^b dx = B(a+1, b+1) = \frac{\Gamma(a+1)\Gamma(b+1)}{\Gamma(a+b+2)} \] 33. \[ \frac{d}{dx} \left( \frac{1}{f(x)} \right) = -\frac{f'(x)}{(f(x))^2} \] 34. \[ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} = \ln 2 \] 35. \[ \frac{1}{2\pi i} \oint_C f(z) dz = 0 \quad \text{if } f(z) \text{ is analytic inside and on } C \] 36. \[ \frac{d}{dx} \left( \arcsin x \right) = \frac{1}{\sqrt{1-x^2}} \] 37. \[ \int_0^\infty e^{-x^2} dx = \frac{\sqrt{\pi}}{2} \] 38. \[ \frac{d}{dx} \left( \arctan x \right) = \frac{1}{1+x^2} \] 39. \[ \sum_{n=0}^{\infty} (-1)^n x^{2n} = \frac{1}{1+x^2} \quad \text{for } |x|<1 \] 40. \[ \int_0^1 \frac{1}{\sqrt{1-x^2}} dx = \arcsin(1) - \arcsin(0) = \frac{\pi}{2} \] 41. \[ \frac{d}{dx} \left( \ln |x| \right) = \frac{1}{x} \] 42. \[ \int_0^\infty e^{-ax^2} dx = \frac{1}{2}\sqrt{\frac{\pi}{a}} \] 43. \[ \frac{d}{dx} \left( \cosh x \right) = \sinh x \] 44. \[ \frac{d}{dx} \left( \sinh x \right) = \cosh x \] 45. \[ \int_0^\infty x^n e^{-x} dx = n! \] 46. \[ \frac{d}{dx} \left( \sec x \right) = \sec x \tan x \] 47. \[ \frac{d}{dx} \left( \csc x \right) = -\csc x \cot x \] 48. \[ \int_0^1 \frac{\ln x}{1-x} dx = -\frac{\pi^2}{6} \] 49. \[ \frac{d}{dx} \left( \cot x \right) = -\csc^2 x \] 50. \[ \int_0^\infty \frac{x^{s-1}}{1+x} dx = \frac{\pi}{\sin(\pi s)} \quad \text{for } 0 < s < 1 \]