一、定义
设函数f(x)f(x)f(x)在区间[a,b][a,b][a,b]上连续，设xxx为区间[a,b][a,b][a,b]上的一点，考察定积分
∫axf(x)dx=∫axf(t)dt
\int _a^xf(x)dx=\int _a^xf(t)dt
∫axf(x)dx=∫axf(t)dt
如果上限xxx在区间[a,b][a,b][a,b]上任意变动，则对于每一个取定的xxx值，定积分∫axf(t)dt\int _a^xf(t)dt∫axf(t)dt都有一个对应值，所以它在区间[a,b][a,b][a,b]上定义了一个函数，记为
Φ(x)=∫axf(t)dt
\Phi(x)=\int _a^xf(t)dt
Φ(x)=∫axf(t)dt
该函数就是积分上限函数。
二、变限积分函数求导公式
如果函数f(x)f(x)f(x)连续，ϕ(x)\phi(x)ϕ(x) 和φ(x)\varphi(x)φ(x)可导，那么变限积分函数的求导公式可表示为
Φ′(x)=ddx∫ϕ(x)φ(x)f(t)dt=f[φ(x)]φ′(x)−f[ϕ(x)]ϕ′(x)
\Phi'(x)=\frac{d}{dx}\int_{\phi(x)}^{\varphi(x)}f(t)dt=f[\varphi(x)]\varphi'(x)-f[\phi(x)]\phi'(x)
Φ′(x)=dxd∫ϕ(x)φ(x)f(t)dt=f[φ(x)]φ′(x)−f[ϕ(x)]ϕ′(x)
[推导过程]
记函数f(x)f(x)f(x)的原函数为F(x)F(x)F(x)，则有
F′(x)=f(x)
F'(x)=f(x)
F′(x)=f(x)
或
∫f(x)dx=F(x)+C
\int f(x)dx=F(x)+C
∫f(x)dx=F(x)+C
则对Φ(x)=∫ϕ(x)φ(x)f(t)dt\Phi(x)=\int_{\phi(x)}^{\varphi(x)}f(t)dtΦ(x)=∫ϕ(x)φ(x)f(t)dt由牛顿-莱布尼茨公式∫abf(x)=F(x)∣ab=F(b)−F(a)\int_a^bf(x)=F(x)|_a^b=F(b)-F(a)∫abf(x)=F(x)∣ab=F(b)−F(a)可得
Φ(x)=∫ϕ(x)φ(x)f(t)dt=F(x)∣ϕ(x)φ(x)=F[φ(x)]−F[ϕ(x)]
\Phi(x)=\int_{\phi(x)}^{\varphi(x)}f(t)dt=F(x)|_{\phi(x)}^{\varphi(x)}=F[\varphi(x)]-F[\phi(x)]
Φ(x)=∫ϕ(x)φ(x)f(t)dt=F(x)∣ϕ(x)φ(x)=F[φ(x)]−F[ϕ(x)]
由函数和的求导法则
[u(x)±v(x)]′=u′(x)±v′(x)
[u(x)\pm v(x)]'=u'(x)\pm v'(x)
[u(x)±v(x)]′=u′(x)±v′(x)
可得
Φ′(x)=ddx∫ϕ(x)φ(x)f(t)dt={F[φ(x)]−F[ϕ(x)]}′={F[φ(x)]}′−{F[ϕ(x)]}′
\Phi^{'}(x)=\frac{d}{dx}\int_{\phi(x)}^{\varphi(x)}f(t)dt=\{F[\varphi(x)]-F[\phi(x)]\}'=\{F[\varphi(x)]\}'-\{F[\phi(x)]\}'
Φ′(x)=dxd∫ϕ(x)φ(x)f(t)dt={F[φ(x)]−F[ϕ(x)]}′={F[φ(x)]}′−{F[ϕ(x)]}′
由复合函数的求导法则
{f[g(x)]}′=f′[g(x)]g′(x)
\{f[g(x)]\}'=f'[g(x)]g'(x)
{f[g(x)]}′=f′[g(x)]g′(x)
可得
Φ′(x)={F[φ(x)]}′−{F[ϕ(x)]}′=F′[φ(x)]φ′(x)−F′[ϕ(x)]ϕ′(x)
\Phi^{'}(x)=\{F[\varphi(x)]\}'-\{F[\phi(x)]\}'=F'[\varphi(x)]\varphi'(x)-F'[\phi(x)]\phi'(x)
Φ′(x)={F[φ(x)]}′−{F[ϕ(x)]}′=F′[φ(x)]φ′(x)−F′[ϕ(x)]ϕ′(x)
由(2)式F′(x)=f(x)F'(x)=f(x)F′(x)=f(x)可知F′[φ(x)]=f[φ(x)]F'[\varphi(x)]=f[\varphi(x)]F′[φ(x)]=f[φ(x)] F′[ϕ(x)]=f[ϕ(x)]F'[\phi(x)]=f[\phi(x)]F′[ϕ(x)]=f[ϕ(x)]，则(8)式可改写为
Φ′(x)=F′[φ(x)]φ′(x)−F′[ϕ(x)]ϕ′(x)=f[φ(x)]φ′(x)−f[ϕ(x)]ϕ′(x)
\Phi^{'}(x)=F'[\varphi(x)]\varphi'(x)-F'[\phi(x)]\phi'(x)=f[\varphi(x)]\varphi'(x)-f[\phi(x)]\phi'(x)
Φ′(x)=F′[φ(x)]φ′(x)−F′[ϕ(x)]ϕ′(x)=f[φ(x)]φ′(x)−f[ϕ(x)]ϕ′(x)
三、定理
定理1 如果函数f(x)f(x)f(x)在区间[a，b][a，b][a，b]上连续，则积分上限函数Φ(x)=∫axf(t)dt\Phi(x)=\int _a^xf(t)dtΦ(x)=∫axf(t)dt在[a，b][a，b][a，b]上具有导数，且导数为：
Φ′(x)=ddx∫axf(t)dt=f(x)
\Phi^{'}(x)=\frac{d}{dx}\int _a^xf(t)dt=f(x)
Φ′(x)=dxd∫axf(t)dt=f(x)
四、应用
求极限
limx→0∫x2xet2dtx
\lim_{x \to 0} \frac{\int_x^{2x}e^{t^2}dt}{x}
x→0limx∫x2xet2dt
令函数f(x)=∫x2xet2dtf(x)=\int_x^{2x}e^{t^2}dtf(x)=∫x2xet2dt，则函数 f(x)f(x)f(x) 在x=0x=0x=0 处连续，运用洛必达法则(L’Hôpital’s rule)则有
limx→0∫x2xet2dtx=limn→0f′(x)x′=limn→0f′(x)
\lim_{x \to 0} \frac{\int_x^{2x}e^{t^2}dt}{x}=\lim_{n \to 0} \frac{f'(x)}{x'}=\lim_{n \to 0} f'(x)
x→0limx∫x2xet2dt=n→0limx′f′(x)=n→0limf′(x)
这是一个典型的变限积分函数的求导，根据变限积分函数求导公式(3)可得
f′(x)=ddx∫x2xet2dt=e(2x)2(2x)′−ex2(x)′=2e4x2−ex2
f'(x)=\frac{d}{dx}\int_x^{2x}e^{t^2}dt=e^{(2x)^2}(2x)'-e^{x^2}(x)'=2e^{4x^2}-e^{x^2}
f′(x)=dxd∫x2xet2dt=e(2x)2(2x)′−ex2(x)′=2e4x2−ex2
则有
limx→0∫x2xet2dtx=limx→02e4x2−ex2=1
\lim_{x \to 0} \frac{\int_x^{2x}e^{t^2}dt}{x}=\lim_{x \to 0}2e^{4x^2}-e^{x^2}=1
x→0limx∫x2xet2dt=x→0lim2e4x2−ex2=1