Use Coupon: CART20 and get 20% off on all online Study Material

Total Price: Rs.

There are no items in this cart.
Continue Shopping
pradyot mayank Grade: 12

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

8 years ago

Answers : (1)

Badiuddin askIITians.ismu Expert
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.


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


In a similar fashion, pursuing \frac{\partial\,\textbf J}{\partial\,b}\, yields


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.\,


Please feel free to post as many doubts on our discussion forum as you can.
 If you find any question Difficult to understand - post it here and we will get you the answer and detailed solution very quickly.
 We are all IITians and here to help you in your IIT JEE  & AIEEE preparation.

 All the best.
Askiitians Experts

8 years ago
Think You Can Provide A Better Answer ?
Answer & Earn Cool Goodies
  • Complete JEE Main/Advanced Course and Test Series
  • OFFERED PRICE: Rs. 15,900
  • View Details

Ask Experts

Have any Question? Ask Experts

Post Question

Answer ‘n’ Earn
Attractive Gift
To Win!!! Click Here for details