The most general form of differentiation under the integral sign states that: if
f(x, t) is a continuous and continuously differentiable (i.e., partial
derivatives exist and are themselves continuous) function and the limits of
integration a(x) and b(x) are continuous and continuously differentiable
functions of x, then \frac{\mathrm{d}}{\mathrm{d} x} \int_{a(x)}^{b(x)} f(x,
t) \mathrm{d} t=f(x, b(x)) \cdot b^{\prime}(x)-f(x, a(x)) \cdot
a^{\prime}(x)+\int_{a(x)}^{b(x)} \frac{\partial}{\partial x} f(x, t) \mathrm{d}
t Problems based on Leibniz rule are often asked in NET/JEE/JAM/NBHM
examinations. Here we give a few of them. (If you find any mistakes with the
answers, kindly let us know!)
1. [NBHM]Evaluate f^{\prime}(3), where f(x)=\int_{-x}^{x} \frac{1-\mathrm{e}^{-x y}}{y} \mathrm{~d} y \quad, \mathrm{x}>0 Ans: \frac{2}{3}\left(e^{9}-e^{-9}\right).
2. [Berkeley Problem]Define F(x)=\int_{\sin x}^{\cos x} e^{\left(t^{2}+x t\right)} d t Compute F^{\prime}(0). Ans: F^{\prime}(0)=\frac{1}{2}(e-3)
3. [JAM]Let a be a non zero real number. Then \lim _{x \rightarrow a} \frac{1}{x^{2}-a^{2}} \int _a^x \sin \left(t^{2}\right) dt equals what?
Ans: \frac{\sin \left(a^{2}\right)}{2 a}.
1. [NBHM]Evaluate f^{\prime}(3), where f(x)=\int_{-x}^{x} \frac{1-\mathrm{e}^{-x y}}{y} \mathrm{~d} y \quad, \mathrm{x}>0 Ans: \frac{2}{3}\left(e^{9}-e^{-9}\right).
2. [Berkeley Problem]Define F(x)=\int_{\sin x}^{\cos x} e^{\left(t^{2}+x t\right)} d t Compute F^{\prime}(0). Ans: F^{\prime}(0)=\frac{1}{2}(e-3)
3. [JAM]Let a be a non zero real number. Then \lim _{x \rightarrow a} \frac{1}{x^{2}-a^{2}} \int _a^x \sin \left(t^{2}\right) dt equals what?
Ans: \frac{\sin \left(a^{2}\right)}{2 a}.
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