ملحق:قائمة تكاملات الدوال الأسية

من ويكيبيديا، الموسوعة الحرة
اذهب إلى: تصفح، ‏ ابحث

هذه قائمة بتكاملات الدوال الأسية:

\int e^{cx}\;\mathrm{d}x = \frac{1}{c} e^{cx}
\int a^{cx}\;\mathrm{d}x = \frac{1}{c\cdot \ln a} a^{cx} for a > 0,\ a \ne 1
\int xe^{cx}\; \mathrm{d}x = \frac{e^{cx}}{c^2}(cx-1)
\int x^2 e^{cx}\;\mathrm{d}x = e^{cx}\left(\frac{x^2}{c}-\frac{2x}{c^2}+\frac{2}{c^3}\right)
\int x^n e^{cx}\; \mathrm{d}x = \frac{1}{c} x^n e^{cx} - \frac{n}{c}\int x^{n-1} e^{cx} \mathrm{d}x
\int\frac{e^{cx}}{x}\; \mathrm{d}x = \ln|x| +\sum_{n=1}^\infty\frac{(cx)^n}{n\cdot n!}
\int\frac{e^{cx}}{x^n}\; \mathrm{d}x = \frac{1}{n-1}\left(-\frac{e^{cx}}{x^{n-1}}+c\int\frac{e^{cx} }{x^{n-1}}\,\mathrm{d}x\right) \qquad\mbox{(for }n\neq 1\mbox{)}
\int e^{cx}\ln x\; \mathrm{d}x = \frac{1}{c}e^{cx}\ln|x|-\operatorname{Ei}\,(cx)
\int e^{cx}\sin bx\; \mathrm{d}x = \frac{e^{cx}}{c^2+b^2}(c\sin bx - b\cos bx)
\int e^{cx}\cos bx\; \mathrm{d}x = \frac{e^{cx}}{c^2+b^2}(c\cos bx + b\sin bx)
\int e^{cx}\sin^n x\; \mathrm{d}x = \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\;\mathrm{d}x
\int e^{cx}\cos^n x\; \mathrm{d}x = \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\;\mathrm{d}x
\int x e^{c x^2 }\; \mathrm{d}x= \frac{1}{2c} \;  e^{c x^2}
\int e^{-c x^2 }\; \mathrm{d}x= \sqrt{\frac{\pi}{4c}} \mbox{erf}(\sqrt{c} x) (\mbox{erf} is the Error function)
\int xe^{-c x^2 }\; \mathrm{d}x=-\frac{1}{2c}e^{-cx^2}
\int {1 \over \sigma\sqrt{2\pi} }\,e^{-{(x-\mu )^2 / 2\sigma^2}}\; \mathrm{d}x= \frac{1}{2} (1 + \mbox{erf}\,\frac{x-\mu}{\sigma \sqrt{2}})
\int e^{x^2}\,\mathrm{d}x = 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}}\;\mathrm{d}x  \quad \mbox{valid for } n > 0,
where  c_{2j}=\frac{ 1 \cdot 3 \cdot 5 \cdots (2j-1)}{2^{j+1}}=\frac{(2j)\,!}{j!\, 2^{2j+1}} \ .