高數微積分公式
常用三角函數
\[\csc{x} = \frac{1}{\sin{x}} \]
\[\sec{x} = \frac{1}{\cos{x}} \]
\[\cot{x} = \frac{1}{\tan{x}} \]
微積分公式
\[\int{tanx}dx = -ln|\cos x|dx + c \]
\[\int \cot{x}dx = \ln{|\sin{x}|}dx + c \]
\[\int{\sec{x}}dx = \ln{|\sec{x}+\tan{x}|dx}+c \]
\[\int{\csc{x}}dx = \ln{|\csc{x}-\cot{x}|dx}+c \]
\[\int{\sec^2{x}}dx = \tan{x}+c \]
\[\int{\csc^2{x}}dx = -\cot{x}+c \]
\[\int\frac{1}{a^2+x^2}dx = \frac{1}{a}\arctan{\frac{x}{a}}+c \]
\[\int\frac{1}{a^2-x^2}dx = \frac{1}{2a}\ln|\frac{a+x}{a-x}|+c \]
\[\int\frac{1}{\sqrt{a^2-x^2}}dx = \arcsin{\frac{x}{a}}+c \]
\[\int\frac{1}{\sqrt{x^2\pm a^2}}dx = \ln{|x+\sqrt{x^2\pm a^2}|}+c \]
\[\int{\sqrt{x^2+a^2}}dx = \frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}\ln{(x+\sqrt{x^2+a^2})}+c \]
\[\int{\sqrt{x^2-a^2}}dx = \frac{x}{2}\sqrt{x^2-a^2}-\frac{a^2}{2}ln{|x+\sqrt{x^2-a^2}|}+c \]
\[\int{\sqrt{a^2-x^2}}dx = \frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\arcsin\frac{x}{a}+c \]
分部積分法
\[\int{u(x)v^{'}(x)}dx = u(x)v(x) - \int{v(x)u^{'}(x)}dx \\等價於 \\ \int{u(x)dv(x)} = u(x)v(x)-\int{v(x)du(x)} \]
華里士公式
\[\int_{0}^{\frac{\pi}{2}}{sin^{n}x}dx = \int_{0}^{\frac{\pi}{2}}{cos^{n}x}dx=\begin{cases} \frac{n-1}{n}\times\frac{n-3}{n-2}\times...\frac{1}{2}\times\frac{\pi}{2},n為偶數\\\frac{n-1}{n}\times\frac{n-3}{n-2}\times...\frac{2}{3}\times1,n為奇數 \end{cases} \]
重要的反常積分
\[\int_{-\infty}^{\infty}{e^{-x^2}}dx = 2\int_{0}^{\infty}{e^{-x^2}}dx = \sqrt{\pi} \]
積分化簡
\[\int{x^n\ln{x}}dx = \frac{1}{n+1}\int{\ln{x}}dx^{n+1} \]