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.

Variable-Limit Integrals

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

Worked Examples

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

Solution: Upper limit $v(x)=x^2$, so

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

Example 2: A Typical Graduate-Entrance-Exam Problem

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 $f(x)$.

Solution:

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. Conclusion: $x=0$ is a local minimum of $f(x)$. This mirrors common exam tasks: differentiate the integral, then use higher derivatives to classify the point.

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Summary

Symbols Used in This Article