Typical Applications

Once you know the definition and derivative property, practice is key. These examples show how “differentiate an upper-limit integral” works in varied settings.

基本应用

例 1：$F(x) = \int_0^x \sin t \, dt$，求 $F'(x)$

解：By $F'(x)=f(x)$, we get $F'(x) = \sin x$.

例 2：$F(x) = \int_1^x \dfrac{1}{t} \, dt$，求 $F'(x)$

解：Treat $f(t)=\frac{1}{t}$, so $F'(x) = \dfrac{1}{x}$.

复杂应用

例 3：$F(x) = \int_0^x e^{-t^2} \, dt$，求 $F'(x)$

解：$F'(x) = e^{-x^2}$. Even without an elementary antiderivative, the derivative is immediate.

例 4：$F(x) = \int_0^x \dfrac{\sin t}{t} \, dt$，求 $F'(x)$

解：$F'(x) = \dfrac{\sin x}{x}$. As long as the integrand is meaningful at the upper limit, the derivative exists.

These examples highlight: the derivative of the upper-limit integral equals the integrand evaluated at the upper limit, regardless of how hard the integral itself is.

总结

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