فرمول های رایج انتگرال (Integral)، در ریاضیات (Mathematics)
\[ \int {k dx} = kx + C \] \[ \int {{x^n}dx} = {{{x^{n + 1}}} \over {n + 1}} + C \ \ \ \ \left( {n \ne - 1} \right) \] \[ \int {{1 \over x}dx} = \ln \left| x \right| + C \] \[ \int {{e^x}dx} = {e^x} + C \] \[ \int {{a^x}dx} = {{{a^x}} \over {\ln a}} + C \ \ \ \ \left( {a > 0 , a \ne 1} \right) \] \[ \int {{b^{ax}}} dx = {1 \over a}{{{b^{ax}}} \over {\ln b}} + C \ \ \ \ ,b > 0,b \ne 1 \] \[ \int {{x^n}} {b^{ax}}dx = {{{x^n}{b^{ax}}} \over {a\ln b}} - {n \over {a\ln b}}\int {{x^{n - 1}}} {b^{ax}}dx \ \ \ \ ,b > 0,b \ne 1 \] \[ \int {\sin x dx} = - \cos x + C \] \[ \int {\cos x dx} = \sin x + C \] \[ \int {{{\sec }^2}x dx} = \tan x + C \] \[ \int {{{\csc }^2}x dx} = - \cot x + C \] \[ \int {\sec x\tan x dx} = \sec x + C \] \[ \int {\csc x\cot x dx} = - \csc x + C \] \[ \int {\tan x dx} = \ln \left| {\sec x} \right| + C \] \[ \int {\cot x dx} = \ln \left| {\sin x} \right| + C \] \[ \int {\sinh x dx} = \cosh x + C \] \[ \int {\cosh x dx} = \sinh x + C \] \[ \int {{1 \over {\sqrt {{a^2} - {x^2}} }}dx} = {\sin ^{ - 1}}{x \over a} + C \] \[ \int {{1 \over {{a^2} + {x^2}}}dx} = {1 \over a}{\tan ^{ - 1}}{x \over a} + C \] \[ \int {{1 \over {x\sqrt {{x^2} - {a^2}} }}dx} = {1 \over a}{\sec ^{ - 1}}\left| {{x \over a}} \right| + C \] \[ \int {{1 \over {\sqrt {{a^2} + {x^2}} }}dx} = {\sinh ^{ - 1}}{x \over a} + C \ \ \ \ \left( {a > 0} \right) \] \[ \int {{1 \over {\sqrt {{x^2} - {a^2}} }}dx} = {\cosh ^{ - 1}}{x \over a} + C \ \ \ \ \left( {x > a > 0} \right) \]
\[ \int {{1 \over {ax + b}}dx} = {1 \over a}\ln \left| {ax + b} \right| + C \] \[ \int {{{{f^\prime }\left( x \right)} \over {f\left( x \right)}}dx} = \ln \left| {f\left( x \right)} \right| + C \] \[ \int {{{\left( {f\left( x \right)} \right)}^n}{f^\prime }\left( x \right)dx} = {{{{\left( {f\left( x \right)} \right)}^{n + 1}}} \over {n + 1}} + C \] \[ \int {f\left( x \right){f^\prime }\left( x \right)dx} = {{{{\left( {f\left( x \right)} \right)}^2}} \over 2} + C \] \[ \int u dv = uv - \int v du \] \[ \int {f\left( x \right){g^\prime }\left( x \right)dx} = f\left( x \right)g\left( x \right) - \int {{f^\prime }\left( x \right)g\left( x \right)dx} \] \[ \int\limits_a^b {f\left( x \right){g^\prime }\left( x \right)dx} = \left[ {f\left( x \right)g\left( x \right)} \right]_{ a}^{ b} - \int\limits_a^b {{f^\prime }\left( x \right)g\left( x \right)dx} \] \[ \int {\left( {f\left( x \right){g^\prime }\left( x \right) + {f^\prime }\left( x \right)g\left( x \right)} \right)dx} = f\left( x \right)g\left( x \right) \] \[ \int\limits_a^b {\left( {f\left( x \right){g^\prime }\left( x \right) + {f^\prime }\left( x \right)g\left( x \right)} \right)dx} = \left[ {f\left( x \right)g\left( x \right)} \right]_{ a}^{ b} \]
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Thomas' Calculus Early Transcendentals - George B. Thomas Jr., Maurice D. Weir, Joel R. Hass - 13th Edition
نویسنده علیرضا گلمکانی
شماره کلید 10084
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