Integral Derivation with Variable Limits

Prerequisite knowledge: Newton-Leibniz formula

introduce

Shaped like

$$G(x)={\int }_{v(x)}^{u(x)}f(t)dt$$

The integral of is called variable limit integral.

According to the Newton-Leibniz formula ,

$$G^{\prime }(x)=F^{\prime }(u(x))u^{\prime }\left(x\right)-F^{\prime }(v(x))v^{\prime }\left(x\right)=f(u(x))u^{\prime }(x)-f(v(x))v^{\prime }\left(x\right)$$

Then we can use this formula to find the derivative of the variable limit integral.

Example 1

令$F(x)={\int }_1^{x^2}{e}^{t}dt$, find$F^{\prime }\left(x\right)$

解：
$F^{\prime }\left(x\right)=e^{x^2}\cdot 2x\_-e^1\cdot 0=2x{e}^{x^2}$

Example 2

计算$\underset{x\to 0}{\lim }\int \frac{{\int }_0^x\sin t^{2}d}{x^2\ln (1+x)}$

Solution:
\qquadReplaced by infinitesimal , the original formula $=\underset{x\to 0}{\lim }\int \frac{{\int }_0^x\sin t^{2}d}{x^2\cdot x}$

\qquad According to L'Hopital's rule , the original formula $=\underset{x\to 0}{\lim }\frac{\sin x^2}{3x{\_}^2}=\underset{x\to 0}{\lim }\frac{x^2}{3x{\_}^2}=\frac{1}{3}$