قائمة تكاملات الدوال الأسية

هذه قائمة خاصة بـتكاملات الدوال الأسية:

$\int e^{cx}\mathrm{d}x=\frac{1}{c}{e}^{cx}$

$\int a^{cx}\mathrm{d}x=\frac{1}{c\cdot \ln a}{a}^{cx}\text{(for }a>0,a\ne 1\text{)}$

$\int x{e}^{cx}\mathrm{d}x=\frac{e^{cx}}{c^2}(cx-1)$

$\int x^2{e}^{cx}\mathrm{d}x=e^{cx}(\frac{x^2}{c}-\frac{2x}{c^2}+\frac{2}{c^3})$

$\int x^n{e}^{cx}\mathrm{d}x=\frac{1}{c}{x}^n{e}^{cx}-\frac{n}{c}\int x^{n-1}{e}^{cx}\mathrm{d}x$

$\int \frac{e^{cx}}{x}\mathrm{d}x=\ln \left|x\right|+\sum _{n=1}^{\mathrm{\infty }}\frac{(cx{)}^n}{n\cdot n!}$

$\int \frac{e^{cx}}{x^n}\mathrm{d}x=\frac{1}{n-1}(-\frac{e^{cx}}{x^{n-1}}+c\int \frac{e^{cx}}{x^{n-1}}\mathrm{d}x)\text{(for }n\ne 1\text{)}$

$\int e^{cx}\ln x\mathrm{d}x=\frac{1}{c}{e}^{cx}\ln \left|x\right|-\mathrm{E}\mathrm{i}(cx)$

$\int e^{cx}\sin bx\mathrm{d}x=\frac{e^{cx}}{c^2+b^2}(c\sin bx-b\cos bx)$

$\int e^{cx}\cos bx\mathrm{d}x=\frac{e^{cx}}{c^2+b^2}(c\cos bx+b\sin bx)$

$\int e^{cx}{\sin }^{n}x\mathrm{d}x=\frac{e^{cx}{\sin }^{n-1}x}{c^2+n^2}(c\sin x-n\cos x)+\frac{n(n-1)}{c^2+n^2}\int e^{cx}{\sin }^{}x\mathrm{d}x$

$\int e^{cx}{\cos }^{n}x\mathrm{d}x=\frac{e^{cx}{\cos }^{n-1}x}{c^2+n^2}(c\cos x+n\sin x)+\frac{n(n-1)}{c^2+n^2}\int e^{cx}{\cos }^{}x\mathrm{d}x$

$\int x{e}^{c{x}^2}\mathrm{d}x=\frac{1}{2c}{e}^{c{x}^2}$

$\int e^{-c{x}^2}\mathrm{d}x=\sqrt{\frac{\pi }{4c}}\text{erf}(\sqrt{c}x)$ ($\text{erf}$ هي دالة الخطأ)

$\int x{e}^{-c{x}^2}\mathrm{d}x=-\frac{1}{2c}{e}^{-c{x}^2}$

$\int \frac{1}{\sigma \sqrt{2\pi }}{e}^{-(x-\mu {)}^2/2{\sigma }^2}\mathrm{d}x=\frac{1}{2}(1+\text{erf}\frac{x-\mu }{\sigma \sqrt{2}})$

$\int e^{x^2}\mathrm{d}x=e^{x^2}\left(\sum _{j=0}^{n-1}{c}_{2j}\frac{1}{x^{2j+1}}\right)+(2n-1)c_{2n-2}\int \frac{e^{x^2}}{x^{2n}}\mathrm{d}x\text{valid for }n>0,$

حيث:

$c_{2j}=\frac{1\cdot 3\cdot 5\cdots (2j-1)}{2^{j+1}}=\frac{(2j)!}{j!2^{2j+1}}.$