✨Danh sách tích phân với hàm mũ

Dưới đây là danh sách các tích phân với hàm mũ.

: $\backslash int\; e^\{cx\}\backslash ;dx\; =\; \backslash frac\{1\}\{c\}\; e^\{cx\}$

: $\backslash int\; a^\{cx\}\backslash ;dx\; =\; \backslash frac\{1\}\{c\; \backslash ln\; a\}\; a^\{cx\}\; \backslash qquad\backslash mbox\{(\}\; a\; >\; 0,\backslash mbox\{\; \}a\; \backslash ne\; 1\backslash mbox\{)\}$

: $\backslash int\; xe^\{cx\}\backslash ;\; dx\; =\; \backslash frac\{e^\{cx\{c^2\}(cx-1)$

: $\backslash int\; x^2\; e^\{cx\}\backslash ;dx\; =\; e^\{cx\}\backslash left(\backslash frac\{x^2\}\{c\}-\backslash frac\{2x\}\{c^2\}+\backslash frac\{2\}\{c^3\}\backslash right)$

: $\backslash int\; x^n\; e^\{cx\}\backslash ;\; dx\; =\; \backslash frac\{1\}\{c\}\; x^n\; e^\{cx\}\; -\; \backslash frac\{n\}\{c\}\backslash int\; x^\{n-1\}\; e^\{cx\}\; dx$

: $\backslash int\backslash frac\{e^\{cx\}\backslash ;\; dx\}\{x\}\; =\; \backslash ln|x|\; +\backslash sum\_\{i=1\}^\backslash infty\backslash frac\{(cx)^i\}\{i\backslash cdot\; i!\}$

: $\backslash int\backslash frac\{e^\{cx\}\backslash ;\; dx\}\{x^n\}\; =\; \backslash frac\{1\}\{n-1\}\backslash left(-\backslash frac\{e^\{cx\{x^\{n-1+c\backslash int\backslash frac\{e^\{cx\}\; \}\{x^\{n-1\backslash ,dx\backslash right)\; \backslash qquad\backslash mbox\{(\}n\backslash neq\; 1\backslash mbox\{)\}$

: $\backslash int\; e^\{cx\}\backslash ln\; x\backslash ;\; dx\; =\; \backslash frac\{1\}\{c\}e^\{cx\}\backslash ln|x|-\backslash operatorname\{Ei\}\backslash ,(cx)$

: $\backslash int\; e^\{cx\}\backslash sin\; bx\backslash ;\; dx\; =\; \backslash frac\{e^\{cx\{c^2+b^2\}(c\backslash sin\; bx\; -\; b\backslash cos\; bx)$

: $\backslash int\; e^\{cx\}\backslash cos\; bx\backslash ;\; dx\; =\; \backslash frac\{e^\{cx\{c^2+b^2\}(c\backslash cos\; bx\; +\; b\backslash sin\; bx)$

: $\backslash int\; e^\{cx\}\backslash sin^n\; x\backslash ;\; dx\; =\; \backslash frac\{e^\{cx\}\backslash sin^\{n-1\}\; x\}\{c^2+n^2\}(c\backslash sin\; x-n\backslash cos\; x)+\backslash frac\{n(n-1)\}\{c^2+n^2\}\backslash int\; e^\{cx\}\backslash sin^\{n-2\}\; x\backslash ;dx$

: $\backslash int\; e^\{cx\}\backslash cos^n\; x\backslash ;\; dx\; =\; \backslash frac\{e^\{cx\}\backslash cos^\{n-1\}\; x\}\{c^2+n^2\}(c\backslash cos\; x+n\backslash sin\; x)+\backslash frac\{n(n-1)\}\{c^2+n^2\}\backslash int\; e^\{cx\}\backslash cos^\{n-2\}\; x\backslash ;dx$

:$\backslash int\; x\; e^\{c\; x^2\; \}\backslash ;\; dx=\; \backslash frac\{1\}\{2c\}\; \backslash ;\; e^\{c\; x^2\}$

:$\backslash int\; \{1\; \backslash over\; \backslash sigma\backslash sqrt\{2\backslash pi\}\; \}\backslash ,e^\{-\{(x-\backslash mu)^2\; /\; 2\backslash sigma^2\backslash ;\; dx=\; \backslash frac\{1\}\{2\; \backslash sigma\}\; (1\; +\; \backslash mbox\{erf\}\backslash ,\backslash frac\{x-\backslash mu\}\{\backslash sigma\; \backslash sqrt\{2)$

:$\backslash int\; e^\{x^2\}\backslash ,dx\; =\; e^\{x^2\}\backslash left(\backslash sum${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 (n > 0), ::với $c${2j}=\frac{ 1 \cdot 3 \cdot 5 \cdots (2j-1)}{2^{j+1=\frac{(2j)\,!}{j!\, 2^{2j+1.

:$\backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; e^\{-ax^2\}\backslash ,dx=\backslash sqrt\{\backslash pi\; \backslash over\; a\}$

:$\backslash int\_\{0\}^\{\backslash infty\}\; x^\{2n\}\; e^\{-\{x^2\}/\{a^2\backslash ,dx=\backslash sqrt\{\backslash pi\}\; \{(2n)!\; \backslash over\; \{n!\; \{\backslash left\; (\backslash frac\{a\}\{2\}\; \backslash right)\}^\{2n\; +\; 1\}$