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}{ an{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}+ an{x}|dx}+c
]
[int{csc{x}}dx = ln{|csc{x}-cot{x}|dx}+c
]
[int{sec^2{x}}dx = an{x}+c
]
[int{csc^2{x}}dx = -cot{x}+c
]
[intfrac{1}{a^2+x^2}dx = frac{1}{a}arctan{frac{x}{a}}+c
]
[intfrac{1}{a^2-x^2}dx = frac{1}{2a}ln|frac{a+x}{a-x}|+c
]
[intfrac{1}{sqrt{a^2-x^2}}dx = arcsin{frac{x}{a}}+c
]
[intfrac{1}{sqrt{x^2pm a^2}}dx = ln{|x+sqrt{x^2pm 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}arcsinfrac{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=egin{cases} frac{n-1}{n} imesfrac{n-3}{n-2} imes...frac{1}{2} imesfrac{pi}{2},n为偶数\frac{n-1}{n} imesfrac{n-3}{n-2} imes...frac{2}{3} imes1,n为奇数 end{cases}
]
重要的反常积分
[int_{-infty}^{infty}{e^{-x^2}}dx = 2int_{0}^{infty}{e^{-x^2}}dx = sqrt{pi}
]
积分化简
[int{x^nln{x}}dx = frac{1}{n+1}int{ln{x}}dx^{n+1}
]