Differentiation of Integral Function
Category : JEE Main & Advanced
If \[{{g}_{1}}(x)\] and \[{{g}_{2}}(x)\] both functions are defined on \[[a,\,\,b]\] and differentiable at a point \[x\in (a,b)\] and \[f(t)\] is continuous for \[{{g}_{1}}(a)\le f(t)\le {{g}_{2}}(b)\], then
\[\frac{d}{dx}\int_{{{g}_{1}}(x)}^{{{g}_{2}}(x)}{f(t)dt}=f[{{g}_{2}}(x)]{{{g}'}_{2}}(x)-f[{{g}_{1}}(x)]{{{g}'}_{1}}(x)\]
\[=f[{{g}_{2}}(x)]\frac{d}{dx}{{g}_{2}}(x)-f[{{g}_{1}}(x)]\frac{d}{dx}{{g}_{1}}(x)\].
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