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:


ANSWER

$$sin\theta+ cosec\theta$$


SOLUTION

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

View Full Answer

Its FREE, you're just one step away


Single Correct Medium Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86
Enroll Now For FREE

Realted Questions

Q1 Subjective Hard
Integrate:
$$\displaystyle \int \cos x \log \cos x\ dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Single Correct Medium
$$\displaystyle \int\frac{dt}{(6t-1)}$$ is equal to:
  • A. $$\ln(6t-1)+C$$
  • B. $$-\dfrac{1}{6}\ln(6t-1)+C$$
  • C. None of these
  • D. $$\dfrac{1}{6} \ln(6t-1) +C$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Subjective Hard
Solve
$$\int { { x }^{ 3 } } \log xdx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 One Word Medium
Find m if :$$\displaystyle \int_{0}^{\pi}\left ( \dfrac{\sqrt{1+\cos\ 2x}}{2} \right )dx=\sqrt m$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Single Correct Medium
$$\displaystyle \int 32x^{3}(\log x)^{2}dx$$ is equal to
  • A. $$8x^{4}(\log x)^{2}+c$$
  • B. $$x^{4}\{8(\log x)^{2}-4\log x\}+c$$
  • C. $$x^{3}\{(\log x)^{2}+2\log x\}+c$$
  • D. $$x^{4}\{8(\log x)^{2}-4\log x+1\}+c$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer