Integration without fun-ction

Calculus Level 3

$\large f\left(\frac1x\right) + x^2 f(x) = 0$

If the function $f(x)$ satisfies the equation above, evalute the integral below.

$\large I = \int_{\sin\theta}^{\csc \theta} f(x) \, dx$

None of these

\pi

\frac { \pi }{ 2 }

1 e 0 Not sufficient information e -1

1 solution

Anupam Khandelwal
Jun 6, 2015

$From\quad the\quad given\quad condition\\ f\left( x \right) =-\frac { 1 }{ { x }^{ 2 } } f\left( \frac { 1 }{ x } \right) \\ I=\int _{ \sin { \theta } }^{ \csc { \theta } }{ -\frac { 1 }{ { x }^{ 2 } } f\left( \frac { 1 }{ x } \right) .dx } \quad \longrightarrow \boxed { 1 } \\ Putting\quad \frac { 1 }{ x } =t\quad \Rightarrow -\frac { 1 }{ { x }^{ 2 } } .dx=dt\\ I=\int _{ \csc { \theta } }^{ \sin { \theta } }{ f(t).dt } =\int _{ \csc { \theta } }^{ \sin { \theta } }{ f(x).dx } \\ From\quad this\quad we\quad get:\\ \quad \quad \quad I=\quad -\int _{ \sin { \theta } }^{ \csc { \theta } }{ f(x).dx } \quad \quad \quad \longrightarrow \boxed { 2 } \\ Adding\quad both\quad equations\quad we\quad get\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \boxed { I=0 }$

Integration without fun-ction

1 solution

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