Variable-limit Integrals and Extension

When the limits of integration are themselves functions of $x$, use the variable-limit differentiation formula—this generalizes the upper-limit integral function.

变限积分

Proof sketch: split the integral into two upper-limit integrals and apply the chain rule.

应用例子

例 1：$F(x) = \int_0^{x^2} \sin t \, dt$，求 $F'(x)$

解：Upper limit $v(x)=x^2$, so

$F'(x) = \sin(x^2) \cdot 2x = 2x \sin(x^2)$

例 2：考研真题应用

Given $f(x) = \int_0^x e^{t^2} \sin t\,dt$, $g(x) = \int_0^x e^{t^2} dt \cdot \sin^2 x$, determine whether $x=0$ is an extremum or inflection point of $f(x)$.

解：

1. Upper-limit derivative
   • $f'(x) = e^{x^2} \sin x$
   • $f''(x) = 2x e^{x^2} \sin x + e^{x^2} \cos x$
2. Check extremum
   • $f'(0) = 0$
   • $f''(0) = 1 > 0$
3. 结论：$x=0$ is a local minimum of $f(x)$. This mirrors common exam tasks: differentiate the integral, then use higher derivatives to classify the point.

总结

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