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![[tex] f(x) = f(x_1) + f'(x_1) \cdot (x-x_1) + \frac {f''(x_1)}{2} \cdot (x-x_1)^2 + \frac{f''' (x_1)}{6} \cdot (x-x_1)^3 + \frac{f^{IV}}{24} (\xi_1) \cdot (x-x_1)^4 dx [/tex]](images/latex/f0c0448d1338fb8a92762dec14ac2539a65a2cee_0.png "[tex] f(x) = f(x_1) + f'(x_1) \cdot (x-x_1) + \frac {f''(x_1)}{2} \cdot (x-x_1)^2 + \frac{f''' (x_1)}{6} \cdot (x-x_1)^3 + \frac{f^{IV}}{24} (\xi_1) \cdot (x-x_1)^4 dx [/tex]")
![[tex] \int_{x_0}^{x_2} f(x) dx = \int_{x_0}^{x_2} f(x_1) + f'(x_1) \cdot (x-x_1) + \frac {f''(x_1)}{2} \cdot (x-x_1)^2 + \frac{f''' (x_1)}{6} \cdot (x-x_1)^3 + \frac{f^{IV}}{24} (\xi_1) \cdot (x-x_1)^4 dx [/tex]](images/latex/9d625b1974b44e333949e1327e21c8e7b338f3dc_0.png "[tex] \int_{x_0}^{x_2} f(x) dx = \int_{x_0}^{x_2} f(x_1) + f'(x_1) \cdot (x-x_1) + \frac {f''(x_1)}{2} \cdot (x-x_1)^2 + \frac{f''' (x_1)}{6} \cdot (x-x_1)^3 + \frac{f^{IV}}{24} (\xi_1) \cdot (x-x_1)^4 dx [/tex]")
![[tex] \int_{x_0}^{x_2} f(x) dx = \left[ f(x_1) \cdot (x-x_1) + \frac{f'(x_1)}{2} \cdot (x-x_1)^2 + \frac {f''(x_1)}{6} \cdot (x-x_1)^3 + \frac{f''' (x_1)}{24} \cdot (x-x_1)^4 + \frac{f^{IV}}{120} (\xi_1) \cdot (x-x_1)^5 \right]_{x_0}^{x_2} [/tex]](images/latex/b4e45831561d4b3bbc684993b113ed3a9e81259b_0.png "[tex] \int_{x_0}^{x_2} f(x) dx = \left[ f(x_1) \cdot (x-x_1) + \frac{f'(x_1)}{2} \cdot (x-x_1)^2 + \frac {f''(x_1)}{6} \cdot (x-x_1)^3 + \frac{f''' (x_1)}{24} \cdot (x-x_1)^4 + \frac{f^{IV}}{120} (\xi_1) \cdot (x-x_1)^5 \right]_{x_0}^{x_2} [/tex]")
![[tex] x_2 - x_1 = h [/tex]](images/latex/ac6b0fdb1b657b9e702f3a5c5288f741216eaa76_0.png "[tex] x_2 - x_1 = h [/tex]")
![[tex] x_0 - x_1 = -h [/tex]](images/latex/41cf4a4845ff416918f5aaf489478573ac5e1f2d_0.png "[tex] x_0 - x_1 = -h [/tex]")
![[tex] (x_2 - x_1)^2 - (x_0 - x_1)^2 =(x_2 - x_1)^4 - (x_0 - x_1)^4 = 0 [/tex]](images/latex/d44815f808bf4d4577d4cb1253331bbc392962f1_0.png "[tex] (x_2 - x_1)^2 - (x_0 - x_1)^2 =(x_2 - x_1)^4 - (x_0 - x_1)^4 = 0 [/tex]")
![[tex] (x_2 - x_1)^3 - (x_0 - x_1)^3 = 2h^3 [/tex]](images/latex/5a063ea5e56db3a78561e9544161f56c0775d2d5_0.png "[tex] (x_2 - x_1)^3 - (x_0 - x_1)^3 = 2h^3 [/tex]")
![[tex] (x_2 - x_1)^5 - (x_0 - x_1)^5 = 2h^5 [/tex]](images/latex/fc851d6a9a94aaf3f1c037c714c08ae649416968_0.png "[tex] (x_2 - x_1)^5 - (x_0 - x_1)^5 = 2h^5 [/tex]")
![[tex] \int_{x_0}^{x_2} f(x) dx = 2hf(x_1) + \frac{h^3 f''(x_1)}{3} + b h^5 f^{IV} (\xi_1) [/tex]](images/latex/3e00f15516c4b3d8f5ead330b022885cb09a42b6_0.png "[tex] \int_{x_0}^{x_2} f(x) dx = 2hf(x_1) + \frac{h^3 f''(x_1)}{3} + b h^5 f^{IV} (\xi_1) [/tex]")
![[tex] f''(x_1) = \frac{ f(x_2) - 2f(x_1) + f(x_0) }{h^2} + b h^2 f^{IV} (\xi_2) [/tex]](images/latex/c063a6b8d552487421cf65fd9711df4f62aed273_0.png "[tex] f''(x_1) = \frac{ f(x_2) - 2f(x_1) + f(x_0) }{h^2} + b h^2 f^{IV} (\xi_2) [/tex]")
![[tex] \int_{x_0}^{x_2} f(x) dx = 2hf(x_1) + \frac{h^3}{3} \left[ \frac{ f(x_2) - 2f(x_1) + f(x_0) }{h^2} + b h^2 f^{IV} (\xi_1) \right] + b h^5 f^{IV} (\xi_2) [/tex]](images/latex/995ca3c89dca1e56a31a1c3b664fa9692480b3c5_0.png "[tex] \int_{x_0}^{x_2} f(x) dx = 2hf(x_1) + \frac{h^3}{3} \left[ \frac{ f(x_2) - 2f(x_1) + f(x_0) }{h^2} + b h^2 f^{IV} (\xi_1) \right] + b h^5 f^{IV} (\xi_2) [/tex]")
![[tex] \int_{x_0}^{x_2} f(x) dx = \frac{h}{3} \left[ f(x_0) + 4 f(x_1) + f(x_2) \right] + c h^5 f^{IV} (\xi_3) [/tex]](images/latex/42b23cf43e27e4f8a670ce73494a770e7fcac55a_0.png "[tex] \int_{x_0}^{x_2} f(x) dx = \frac{h}{3} \left[ f(x_0) + 4 f(x_1) + f(x_2) \right] + c h^5 f^{IV} (\xi_3) [/tex]")
![[tex] f(x) = f(x_0) + f'(x_0) \cdot (x-x_0) + f''(\xi) \cdot \frac {(x-x_0)^2}{2} [/tex]](images/latex/05fddebd6334d51d70612f7e9042120a2bb3f6cc_0.png "[tex] f(x) = f(x_0) + f'(x_0) \cdot (x-x_0) + f''(\xi) \cdot \frac {(x-x_0)^2}{2} [/tex]")
![[tex] \int_{x_0}^{x_1} f(x) dx = \int_{x_0}^{x_1} f(x_0) + f'(x_0) \cdot (x-x_0) + f''(\xi) \cdot \frac {(x-x_0)^2}{2} [/tex]](images/latex/82178926659b12d070a9546e03fe8e4703a76158_0.png "[tex] \int_{x_0}^{x_1} f(x) dx = \int_{x_0}^{x_1} f(x_0) + f'(x_0) \cdot (x-x_0) + f''(\xi) \cdot \frac {(x-x_0)^2}{2} [/tex]")
![[tex] \int_{x_0}^{x_1} f(x) dx = \left[ f(x_0) \cdot (x-x_0) + f'(x_0) \cdot \frac{(x-x_0)^2}{2} + f''(\xi) \cdot \frac{(x-x_0)^3}{6} \right]_{x_0}^{x_1} [/tex]](images/latex/f9e07c789b2af14e29c76a0e401bb434dc80865d_0.png "[tex] \int_{x_0}^{x_1} f(x) dx = \left[ f(x_0) \cdot (x-x_0) + f'(x_0) \cdot \frac{(x-x_0)^2}{2} + f''(\xi) \cdot \frac{(x-x_0)^3}{6} \right]_{x_0}^{x_1} [/tex]")
![[tex] \int_{x_0}^{x_1} f(x) dx = h \cdot f(x_0) + \frac{h^2}{2} \cdot f'(x_0) + \frac{h^3}{6} \cdot f''(\xi) [/tex]](images/latex/82e2ec3c3c557614513149051fba14d223d85fe2_0.png "[tex] \int_{x_0}^{x_1} f(x) dx = h \cdot f(x_0) + \frac{h^2}{2} \cdot f'(x_0) + \frac{h^3}{6} \cdot f''(\xi) [/tex]")
![[tex] f'(x_0) = \frac{ f(x_1) - f(x_0) }{h} [/tex]](images/latex/28c3fcbde4675fe2f17b21bc51f6517d96413f27_0.png "[tex] f'(x_0) = \frac{ f(x_1) - f(x_0) }{h} [/tex]")
![[tex] \int_{x_0}^{x_1} f(x) dx = h f(x_0) + \frac{h}{2} \left( f(x_1) - f(x_0) \right) + f''(\xi) \cdot \frac{h^3}{6} [/tex]](images/latex/0e73bbff1d7cb6dd3f2fe9dfa78cce3a30738946_0.png "[tex] \int_{x_0}^{x_1} f(x) dx = h f(x_0) + \frac{h}{2} \left( f(x_1) - f(x_0) \right) + f''(\xi) \cdot \frac{h^3}{6} [/tex]")
![[tex] \int_{x_0}^{x_1} f(x) dx = \frac{h}{2} \left[ f(x_1) + f(x_0) \right] + f''(\xi) \cdot \frac{h^3}{6} [/tex]](images/latex/de70c1c8427864d573c14c031194836907562ca4_0.png "[tex] \int_{x_0}^{x_1} f(x) dx = \frac{h}{2} \left[ f(x_1) + f(x_0) \right] + f''(\xi) \cdot \frac{h^3}{6} [/tex]")
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