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        what is newton -leibnitz's rule how to apply it in a question ?explain with 3-4 example
8 years ago

147 Points
										Dear pradyot
Leibniz Integral Rule
The Leibniz integral rule gives a formula for differentiation of a definite integral whose limits are functions of the differential  variable,

where the partial derivative of f indicates that inside the integral only the variation of ƒ ( x, α ) with α is considered in taking the derivative.
Example
Here, we consider the integration of
$\textbf I\;=\;\int_0^{\frac{\pi}{2}}\,\frac{1}{\left(a\,\cos^2\,x+b\,\sin^2\,x\right)^2}\;dx,\,$
where both $a,\,b\,>\,0$, by differentiating under the integral sign.
Let us first find $\textbf J\;=\;\int_0^{\frac{\pi}{2}}\,\frac{1}{a\,\cos^2\,x+b\,\sin^2\,x}\;dx.\,$
Dividing both the numerator and the denominator by $\cos^2\,x$ yields
\begin{align} \textbf J\; &=\;\int_0^{\frac{\pi}{2}}\,\frac{\sec^2\,x}{a\,+b\,\tan^2\,x}\;dx \\ &=\,\frac{1}{b}\,\int_0^{\frac{\pi}{2}}\,\frac{1}{\left(\sqrt{\,\frac{a}{b}\,}\right)^2+\tan^2\,x}\;d(\tan\,x)\, \\ &=\,\frac{1}{\sqrt{\,a\,b\,}}\,\left(\tan^{-1}\left(\sqrt{\,\frac{b}{a}\,}\,\tan\,x\right)\right)\,\bigg|_0^{\frac{\pi}{2}}\;=\;\frac{\pi}{2\,\sqrt{\,a\,b\,}}. \end{align}
The limits of integration being independent of $a,\,$ $\textbf J\;=\;\int_0^{\frac{\pi}{2}}\,\frac{1}{a\,\cos^2\,x+b\,\sin^2\,x}\;dx\,$ gives us
$\frac{\partial\,\textbf J}{\partial\,a}\;=\;-\,\int_0^{\frac{\pi}{2}}\,\frac{\cos^2\,x\;dx}{\left(a\,\cos^2\,x+b\,\sin^2\,x\right)^2}\,$
whereas $\textbf J\;=\;\frac{\pi}{2\,\sqrt{\,a\,b\,}}$ gives us
$\frac{\partial\,\textbf J}{\partial\,a}\;=\;-\frac{\pi}{4\,\sqrt{\,a^3\,b\,}}.\,$
Equating these two relations then yields
$\,\int_0^{\frac{\pi}{2}}\,\frac{\cos^2\,x\;dx}{\left(a\,\cos^2\,x+b\,\sin^2\,x\right)^2}\;=\;\frac{\pi}{4\,\sqrt{\,a^3\,b\,}}.\,$
In a similar fashion, pursuing $\frac{\partial\,\textbf J}{\partial\,b}\,$ yields
$\,\int_0^{\frac{\pi}{2}}\,\frac{\sin^2\,x\;dx}{\left(a\,\cos^2\,x+b\,\sin^2\,x\right)^2}\;=\;\frac{\pi}{4\,\sqrt{\,a\,b^3\,}}.\,$
Adding the two results then produces
$\textbf I\;=\;\int_0^{\frac{\pi}{2}}\,\frac{1}{\left(a\,\cos^2\,x+b\,\sin^2\,x\right)^2}\;dx\;=\;\frac{\pi}{4\,\sqrt{\,a\,b\,}}\left(\frac{1}{a}+\frac{1}{b}\right),\,$
which is the value of the integral $\textbf I.\,$

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8 years ago
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