Mathematics

# If f(x) is a function satisfying $f\left(\frac{1}{x}\right)+x^2f(x) =0$ for all non-zero x, then $\int_\limits{sin\theta}^{cosec\theta}f(x)dx$ equals to:

$sin\theta+ cosec\theta$

##### SOLUTION

$let\>x=(\frac{1}{t})then\>dx\>=(\frac{-1}{t^2})dt\>\\\therefore\>x=sin\theta\>,\>then\>t=cosec\theta\>\\\>and\>x=\>cosec\theta\>,then\>t=\>sin\>\theta\>\\I\>=\int_{cosec\theta\>}^{sin\theta\>}f((\frac{1}{t}))(\frac{-1}{t^2})dt\\=\int_{cosec\theta\>}^{sin\theta\>}\>-t^2\>f(t)\>(\frac{-1}{t^2})dt\\=\>\int_{cosec\theta\>}^{sin\theta\>}f(t)dt=\int_{cosec\theta\>}^{sin\theta\>}f(x)dx\\\>\therefore\>2I=0\>then\>I\>=\>0$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

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