常用积分、极限表

本文给出基本积分表和常用积分表。

基本积分表

$$ \begin{array}{c} \int x^{a} \mathrm{~d} x=\frac{1}{a+1} x^{a+1}+C \quad(a \in R, a \neq-1)\\ \int \frac{1}{x} \mathrm{~d} x=\ln |x|+C\\ \int \cos x \mathrm{~d} x=\sin x+C\\ \int \sin x \mathrm{~d} x=-\cos x+C\\ \int \tan x \mathrm{~d} x=-\ln |\cos x|+C\\ \int \cot x \mathrm{~d} x=\ln |\sin x|+C\\ \int \frac{1}{\cos ^{2} x} \mathrm{~d} x=\tan x+C \\ \int \frac{1}{1+x^{2}} \mathrm{~d} x=\arctan x+C \\ \int \mathrm{e}^{x} \mathrm{~d} x=\mathrm{e}^{x}+C\\ \int x \mathrm{e}^{x} \mathrm{~d} x=\mathrm{e}^{x}(x-1)+C \\ \int a^{x} \mathrm{~d} x=\frac{1}{\ln a} a^{x}+C \end{array} $$

常用积分表

$$ \begin{array}{l}\int \sin ^{2} x \mathrm{~d} x=\frac{1}{2}(x-\sin x \cos x)+C \\ \int \cos ^{2} x \mathrm{~d} x=\frac{1}{2}(x+\sin x \cos x)+C \\ \int \sec x \mathrm{~d} x=\ln |\tan x+\sec x|+C\\ \int \ln x \mathrm{~d} x=x \ln x-x+C \\ \int \sqrt{a^{2}-x^{2}} \mathrm{~d} x=\frac{1}{2}\left(x \sqrt{a^{2}-x^{2}}+a^{2} \arcsin \frac{x}{a}\right)+C \\ \int \frac{1}{\sqrt{1-x^{2}}} \mathrm{~d} x=\arcsin (x)+C \\ \int \frac{1}{\sqrt{1+x^{2}}} \mathrm{~d} x=\ln \left(x+\sqrt{1+x^{2}}\right)+C=\sinh ^{-1} x+C \\ \int_{-\infty}^{\infty} e^{-x^{2}} d x=\sqrt{\pi} \end{array} $$

定积分

$$
\int_{0}^{\pi / 2} \cos ^{n} \theta \mathrm{d} \theta=\frac{\sqrt{\pi}}{2} \frac{\Gamma(n / 2+1 / 2)}{\Gamma(n / 2+1)}
$$

常用极限

$$
\begin{array}{l}
\lim _{x \rightarrow 0} \frac{\sin x}{x}=1 \
\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{x}=e \

\end{array}
$$

参考资料

• https://wuli.wiki/changed/ITable.html#eq13

文章链接：
https://www.zywvvd.com/notes/study/math/calculus/integral-table/integral-table/