变限积分函数的求导

设函数 f ( x ) f(x) f(x)在区间 [ a , b ] [a,b] [a,b]上连续，设 x x x为区间 [ a , b ] [a,b] [a,b]上的一点，考察定积分 ∫ a x f ( x ) d x = ∫ a x f ( t ) d t \int _a^xf(x)dx=\int _a^xf(t)dt ∫ax​f(x)dx=∫ax​f(t)dt 如果上限 x x x在区间 [ a , b ] [a,b] [a,b]上任意变动，则对于每一个取定的 x x x值，定积分 ∫ a x f ( t ) d t \int _a^xf(t)dt ∫ax​f(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 ) = ∫ a x f ( t ) d t \Phi(x)=\int _a^xf(t)dt Φ(x)=∫ax​f(t)dt 该函数就是积分上限函数。

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 ) = d d x ∫ ϕ ( x ) φ ( x ) f ( t ) d t = 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 ) d x = 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 ) d t \Phi(x)=\int_{\phi(x)}^{\varphi(x)}f(t)dt Φ(x)=∫ϕ(x)φ(x)​f(t)dt由牛顿-莱布尼茨公式 ∫ a b f ( x ) = F ( x ) ∣ a b = F ( b ) − F ( a ) \int_a^bf(x)=F(x)|_a^b=F(b)-F(a) ∫ab​f(x)=F(x)∣ab​=F(b)−F(a)可得 Φ ( x ) = ∫ ϕ ( x ) φ ( x ) f ( t ) d t = 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 ) = d d x ∫ ϕ ( x ) φ ( x ) f ( t ) d t = { 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)

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 ) = ∫ a x f ( t ) d t \Phi(x)=\int _a^xf(t)dt Φ(x)=∫ax​f(t)dt在 [ a ， b ] [a，b] [a，b]上具有导数，且导数为： Φ ′ ( x ) = d d x ∫ a x f ( t ) d t = f ( x ) \Phi^{'}(x)=\frac{d}{dx}\int _a^xf(t)dt=f(x) Φ′(x)=dxd​∫ax​f(t)dt=f(x)

求极限 lim ⁡ x → 0 ∫ x 2 x e t 2 d t x \lim_{x \to 0} \frac{\int_x^{2x}e^{t^2}dt}{x} x→0lim​x∫x2x​et2dt​ 令函数 f ( x ) = ∫ x 2 x e t 2 d t f(x)=\int_x^{2x}e^{t^2}dt f(x)=∫x2x​et2dt，则函数 f ( x ) f(x) f(x) 在 x = 0 x=0 x=0 处连续，运用洛必达法则(L’Hôpital’s rule)则有 lim ⁡ x → 0 ∫ x 2 x e t 2 d t x = lim ⁡ n → 0 f ′ ( x ) x ′ = lim ⁡ n → 0 f ′ ( 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→0lim​x∫x2x​et2dt​=n→0lim​x′f′(x)​=n→0lim​f′(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 ) = d d x ∫ x 2 x e t 2 d t = e ( 2 x ) 2 ( 2 x ) ′ − e x 2 ( x ) ′ = 2 e 4 x 2 − e x 2 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​∫x2x​et2dt=e(2x)2(2x)′−ex2(x)′=2e4x2−ex2

则有 lim ⁡ x → 0 ∫ x 2 x e t 2 d t x = lim ⁡ x → 0 2 e 4 x 2 − e x 2 = 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→0lim​x∫x2x​et2dt​=x→0lim​2e4x2−ex2=1