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