فرمول های انتگرال (Integral) شامل e (نمایی - Exponential)، در ریاضیات (Mathematics)
\[ \int {{e^x}dx} = {e^x} \] \[ \int e^{ax}\ dx = \frac{1}{a}e^{ax} \] \[ \int \sqrt{x} e^{ax}\ dx = \frac{1}{a}\sqrt{x}e^{ax} +\frac{i\sqrt{\pi}}{2a^{3/2}} \text{erf}\left(i\sqrt{ax}\right), \text{ where erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}dt \] \[ \int x e^x\ dx = (x-1) e^x \] \[ \int x e^{ax}\ dx = \left(\frac{x}{a}-\frac{1}{a^2}\right) e^{ax} \] \[ \int x^2 e^{x}\ dx = \left(x^2 - 2x + 2\right) e^{x} \] \[ \int x^2 e^{ax}\ dx = \left(\frac{x^2}{a}-\frac{2x}{a^2}+\frac{2}{a^3}\right) e^{ax} \] \[ \int x^3 e^{x}\ dx = \left(x^3-3x^2 + 6x - 6\right) e^{x} \] \[ \int x^n e^{ax}\ dx = \dfrac{x^n e^{ax}}{a} - \dfrac{n}{a}\int x^{n-1}e^{ax}\hspace{1pt}\text{d}x \] \[ \int x^n e^{ax}\ dx = \frac{(-1)^n}{a^{n+1}}\Gamma[1+n,-ax], \text{ where } \Gamma(a,x)=\int_x^{\infty} t^{a-1}e^{-t}\hspace{2pt}\text{d}t \] \[ \int e^{ax^2}\ dx = -\frac{i\sqrt{\pi}}{2\sqrt{a}}\text{erf}\left(ix\sqrt{a}\right) \] \[ \int e^{-ax^2}\ dx = \frac{\sqrt{\pi}}{2\sqrt{a}}\text{erf}\left(x\sqrt{a}\right) \] \[ \int x e^{-ax^2}\ {dx} = -\dfrac{1}{2a}e^{-ax^2} \] \[ \int x^2 e^{-ax^2}\ {dx} = \dfrac{1}{4}\sqrt{\dfrac{\pi}{a^3}}\text{erf}(x\sqrt{a}) -\dfrac{x}{2a}e^{-ax^2} \] \[ \int {{e^{ax}}} \sin bxdx = {{{e^{ax}}} \over {{a^2} + {b^2}}}(a\;\sin bx - b\cos bx) + C \] \[ \int {{e^{ax}}} \cos bxdx = {{{e^{ax}}} \over {{a^2} + {b^2}}}(a\cos bx + b\sin bx) + C \] \[ \int {{e^{ax}}} \sinh bxdx = {{{e^{ax}}} \over 2}\left[ {{{{e^{bx}}} \over {a + b}} - {{{e^{ - bx}}} \over {a - b}}} \right] + C \ \ \ \ ,{a^2} \ne {b^2} \] \[ \int {{e^{ax}}} \cosh bxdx = {{{e^{ax}}} \over 2}\left[ {{{{e^{bx}}} \over {a + b}} + {{{e^{ - bx}}} \over {a - b}}} \right] + C \ \ \ \ ,{a^2} \ne {b^2} \]
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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 10089
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