||V{maketoc}||
^@#16:
! جدول انتگرال توابع نمایی:
@@{TEX()} {\int e^x\,dx = e^x + C} {TEX}@@
@@{TEX()} {\int a^x\,dx = \frac{a^x}{\ln{a}} + C} {TEX}@@
---
@@{TEX()} {\int e^{cx}\;dx = \frac{1}{c} e^{cx}} {TEX}@@
@@{TEX()} {\int a^{cx}\;dx = \frac{1}{c \ln a} a^{cx} \qquad\mbox{( } a > 0,\mbox{ }a \ne 1\mbox{)}} {TEX}@@
@@{TEX()} {\int xe^{cx}\; dx = \frac{e^{cx}}{c^2}(cx-1)} {TEX}@@
@@{TEX()} {\int x^2 e^{cx}\;dx = e^{cx}\left(\frac{x^2}{c}-\frac{2x}{c^2}+\frac{2}{c^3}\right)} {TEX}@@
@@{TEX()} {\int x^n e^{cx}\; dx = \frac{1}{c} x^n e^{cx} - \frac{n}{c}\int x^{n-1} e^{cx} dx} {TEX}@@
@@{TEX()} {\int\frac{e^{cx}\; dx}{x} = \ln|x| +\sum_{i=1}^\infty\frac{(cx)^i}{i\cdot i!}} {TEX}@@
@@{TEX()} {\int\frac{e^{cx}\; dx}{x^n} = \frac{1}{n-1}\left(-\frac{e^{cx}}{x^{n-1}}+c\int\frac{e^{cx} }{x^{n-1}}\,dx\right) \qquad\mbox{( }n\neq 1\mbox{)}} {TEX}@@
@@{TEX()} {\int e^{cx}\ln x\; dx = \frac{1}{c}e^{cx}\ln|x|-\operatorname{Ei}\,(cx)} {TEX}@@
@@{TEX()} {\int e^{cx}\sin bx\; dx = \frac{e^{cx}}{c^2+b^2}(c\sin bx - b\cos bx)} {TEX}@@
@@{TEX()} {\int e^{cx}\cos bx\; dx = \frac{e^{cx}}{c^2+b^2}(c\cos bx + b\sin bx)} {TEX}@@
@@{TEX()} {\int e^{cx}\sin^n x\; dx = \frac{e^{cx}\sin^{n-1} x}{c^2+n^2}(c\sin x-n\cos x)+\frac{n(n-1)}{c^2+n^2}\int e^{cx}\sin^{n-2} x\;dx} {TEX}@@
@@{TEX()} {\int e^{cx}\cos^n x\; dx = \frac{e^{cx}\cos^{n-1} x}{c^2+n^2}(c\cos x+n\sin x)+\frac{n(n-1)}{c^2+n^2}\int e^{cx}\cos^{n-2} x\;dx} {TEX}@@
@@{TEX()} {\int x e^{c x^2 }\; dx= \frac{1}{2c} \; e^{c x^2}} {TEX}@@
@@{TEX()} {\int {1 \over \sigma\sqrt{2\pi} }\,e^{-{(x-\mu )^2 / 2\sigma^2}}\; dx= \frac{1}{2 \sigma} (1 + \mbox{erf}\,\frac{x-\mu}{\sigma \sqrt{2}})} {TEX}@@
@@{TEX()} {\int e^{x^2}\,dx = e^{x^2}\left( \sum_{j=0}^{n-1}c_{2j}\,\frac{1}{x^{2j+1}} \right )+(2n-1)c_{2n-2} \int \frac{e^{x^2}}{x^{2n}}\;dx \quad \mbox{ } n > 0, } {TEX}@@
بطوریکه:
@@{TEX()} { c_{2j}=\frac{ 1 \cdot 3 \cdot 5 \cdots (2j-1)}{2^{j+1}}=\frac{2j\,!}{j!\, 2^{2j+1}} \ . } {TEX}@@
---
!همچنین ببینید:
*((جدول انتگرال توابع گنگ))
*((جدول انتگرال توابع لگاریتمی))
*((جدول انتگرال توابع نمایی))
*((جدول انتگرال توابع مثلثاتی))
*((جدول انتگرال توابع هیپربولیک))
#@^