高等数学：变限积分与含参积分的两把快刀
变限积分和含参积分是高等数学积分学的高级内容，掌握它们的求导和处理技巧是考研数学的必备能力。
一、变限积分求导公式（莱布尼茨公式）
$\dfrac{d}{dx}\int_{a(x)}^{b(x)} f(t)\,dt = f(b(x))\cdot b'(x) - f(a(x))\cdot a'(x)$文章源自公式库网-https://www.gongshiku.com/html/202207/gaodengshuxuejiqiaochulibianxianjifenyuhancanjifendeliangbakuaidao.html
特别地：$\dfrac{d}{dx}\int_{a}^{x} f(t)\,dt = f(x)$（上限求导）文章源自公式库网-https://www.gongshiku.com/html/202207/gaodengshuxuejiqiaochulibianxianjifenyuhancanjifendeliangbakuaidao.html
$\dfrac{d}{dx}\int_{x}^{b} f(t)\,dt = -f(x)$（下限求导）文章源自公式库网-https://www.gongshiku.com/html/202207/gaodengshuxuejiqiaochulibianxianjifenyuhancanjifendeliangbakuaidao.html
二、含参积分的求导
若被积函数中含参数 $x$：文章源自公式库网-https://www.gongshiku.com/html/202207/gaodengshuxuejiqiaochulibianxianjifenyuhancanjifendeliangbakuaidao.html
$\dfrac{d}{dx}\int_{a}^{b} f(x,t)\,dt = \int_{a}^{b} \dfrac{\partial}{\partial x}f(x,t)\,dt$文章源自公式库网-https://www.gongshiku.com/html/202207/gaodengshuxuejiqiaochulibianxianjifenyuhancanjifendeliangbakuaidao.html
使用条件：$f(x,t)$ 和 $f_x(x,t)$ 在积分区域上连续。文章源自公式库网-https://www.gongshiku.com/html/202207/gaodengshuxuejiqiaochulibianxianjifenyuhancanjifendeliangbakuaidao.html
三、积分次序交换
若 $f(x,y)$ 连续，则二重积分与累次积分的次序可交换：文章源自公式库网-https://www.gongshiku.com/html/202207/gaodengshuxuejiqiaochulibianxianjifenyuhancanjifendeliangbakuaidao.html
$\int_{a}^{b}\int_{c}^{d} f(x,y)\,dy\,dx = \int_{c}^{d}\int_{a}^{b} f(x,y)\,dx\,dy$文章源自公式库网-https://www.gongshiku.com/html/202207/gaodengshuxuejiqiaochulibianxianjifenyuhancanjifendeliangbakuaidao.html
四、典型应用
例1：求 $\dfrac{d}{dx}\int_{0}^{x^2} \sin t^2\,dt$文章源自公式库网-https://www.gongshiku.com/html/202207/gaodengshuxuejiqiaochulibianxianjifenyuhancanjifendeliangbakuaidao.html
解：$= \sin(x^2)^2 \cdot 2x = 2x\sin(x^4)$文章源自公式库网-https://www.gongshiku.com/html/202207/gaodengshuxuejiqiaochulibianxianjifenyuhancanjifendeliangbakuaidao.html
例2：计算 $\int_{0}^{\infty} \dfrac{\arctan(bx) - \arctan(ax)}{x}\,dx$（$a,b > 0$）
解：令 $F(y) = \int_{0}^{\infty} \dfrac{\arctan(yx)}{x}\,dx$，则 $F'(y) = \dfrac{\pi}{2y}$，积分得 $F(b)-F(a) = \dfrac{\pi}{2}\ln\dfrac{b}{a}$。
