高等数学-积分公式
高数微积分公式
常用三角函数
\[\csc{x} = \frac{1}{\sin{x}}
\]
\[\sec{x} = \frac{1}{\cos{x}}
\]
\[\cot{x} = \frac{1}{\tan{x}}
\]
微积分公式
\[\int{tanx}dx = -ln|\cos x|dx + c
\]
\[\int \cot{x}dx = \ln{|\sin{x}|}dx + c
\]
\[\int{\sec{x}}dx = \ln{|\sec{x}+\tan{x}|dx}+c
\]
\[\int{\csc{x}}dx = \ln{|\csc{x}-\cot{x}|dx}+c
\]
\[\int{\sec^2{x}}dx = \tan{x}+c
\]
\[\int{\csc^2{x}}dx = -\cot{x}+c
\]
\[\int\frac{1}{a^2+x^2}dx = \frac{1}{a}\arctan{\frac{x}{a}}+c
\]
\[\int\frac{1}{a^2-x^2}dx = \frac{1}{2a}\ln|\frac{a+x}{a-x}|+c
\]
\[\int\frac{1}{\sqrt{a^2-x^2}}dx = \arcsin{\frac{x}{a}}+c
\]
\[\int\frac{1}{\sqrt{x^2\pm a^2}}dx = \ln{|x+\sqrt{x^2\pm a^2}|}+c
\]
\[\int{\sqrt{x^2+a^2}}dx = \frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}\ln{(x+\sqrt{x^2+a^2})}+c
\]
\[\int{\sqrt{x^2-a^2}}dx = \frac{x}{2}\sqrt{x^2-a^2}-\frac{a^2}{2}ln{|x+\sqrt{x^2-a^2}|}+c
\]
\[\int{\sqrt{a^2-x^2}}dx = \frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\arcsin\frac{x}{a}+c
\]
分部积分法
\[\int{u(x)v^{'}(x)}dx = u(x)v(x) - \int{v(x)u^{'}(x)}dx
\\等价于
\\
\int{u(x)dv(x)} = u(x)v(x)-\int{v(x)du(x)}
\]
华里士公式
\[\int_{0}^{\frac{\pi}{2}}{sin^{n}x}dx = \int_{0}^{\frac{\pi}{2}}{cos^{n}x}dx=\begin{cases} \frac{n-1}{n}\times\frac{n-3}{n-2}\times...\frac{1}{2}\times\frac{\pi}{2},n为偶数\\\frac{n-1}{n}\times\frac{n-3}{n-2}\times...\frac{2}{3}\times1,n为奇数 \end{cases}
\]
重要的反常积分
\[\int_{-\infty}^{\infty}{e^{-x^2}}dx = 2\int_{0}^{\infty}{e^{-x^2}}dx = \sqrt{\pi}
\]
积分化简
\[\int{x^n\ln{x}}dx = \frac{1}{n+1}\int{\ln{x}}dx^{n+1}
\]
