常用积分表¶
\[\int 0dx=C\]
\[\int x^{\alpha}dx=\frac{1}{\alpha+1}x^{\alpha+1}+C\]
\[\int a^{x}dx=\frac{1}{\ln a}a^{x}+C\]
\[\int \frac{1}{x}dx=\ln x+C\]
\[\begin{aligned}\int \sin xdx&=-\cos x+C \quad \int \cos xdx=\sin x+C\\\int \sec^{2}xdx&=\tan x+C \quad \int \csc^{2}xdx=\cot x+C\end{aligned}\]
\[\int \frac{1}{a^2+x^2}dx=\frac{1}{a}\arctan \frac{x}{a}+C\]
\[\int \frac{1}{x^2-a^2}dx=\frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right|+C\]
\[\int \frac{1}{\sqrt{x^2+a^2}}dx=\ln\left|x+\sqrt{x^2+a^2}\right|+C\]
\[\int \frac{1}{\sqrt{x^2-a^2}}dx=\ln\left|x+\sqrt{x^2-a^2}\right|+C\]
\[\int \frac{1}{\sqrt{a^2-x^2}}dx=\arcsin\left(\frac{x}{a}\right)+C\]
\[\int \sqrt{x^2+a^2}dx=\frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}\ln\left|x+\sqrt{x^2+a^2}\right|+C\]
\[\int \sqrt{x^2-a^2}dx=\frac{x}{2}\sqrt{x^2-a^2}-\frac{a^2}{2}\ln\left|x+\sqrt{x^2-a^2}\right|+C\]
\[\int \sqrt{a^2-x^2}dx=\frac{a^2}{2}\arcsin\left(\frac{x}{a}\right)+\frac{x}{2}\sqrt{a^2-x^2}+C\]