Let $$f(x)$$ be an even function such that ∫$$0$$∞ $$f(x)dx=$$π$$2$$, then the value of the definite integral ∫$$0$$∞ fx−$$1xdx$$ is equal to

Question

Let $$f(x)$$ be an even function such that ∫$$0$$∞ $$f(x)dx=$$π$$2$$, then the value of  the definite integral ∫$$0$$∞ fx−$$1xdx$$ is equal to

Infinity Learn · Accepted Answer

Let $$I=$$∫$$0$$∞ fx−$$1xdx$$…………..(1) Put $$x=1t$$⇒$$dx=$$−$$1t2dtFrom$$⁡(1):$$I=$$∫∞$$0$$ $$f1t$$−t−$$dtt2$$  Use ∫ab $$f(x)dx=$$−∫ba $$f(x)dx$$⇒$$I=$$∫$$0$$∞ $$f1t$$−$$tdtt2$$⇒$$I=$$∫$$0$$∞ ft−$$1tdtt2$$ $$($$∵f is even function $$)$$⇒$$I=$$∫$$0$$∞ fx−$$1xdxx2$$…………...(2)Adding$$ equations $$(1) & (2) ⇒$$2I=$$∫$$0$$∞ fx−$$1xdx+$$∫$$0$$∞ fx−$$1xdxx2$$⇒$$2I=$$∫$$0$$∞ fx−$$1x1+1x2dx$$ Put x−$$1x=u$$⇒$$1+1x2dx=du$$∴$$2I=$$∫−∞∞ $$f(u)du$$ ∵ If f is even ∫−aa $$f(x)=2$$×∫$$0a$$ $$f(x)dx=2$$∫$$0$$∞ $$f(x)dx=2$$$$π$$$$2$$⇒$$2I=$$π∴$$I=$$π$$2$$

$Let f (x) be an even function such that \int_{0}^{\infty} f (x) d x = \frac{π}{2}, then the value of$ $the definite integral \int_{0}^{\infty} f (x - \frac{1}{x}) d x is equal to$

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Let f(x) be an even function such that ∫0∞ f(x)dx=π2, then the value of the definite integral ∫0∞ fx−1xdx is equal to

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$Let f (x) be an even function such that \int_{0}^{\infty} f (x) d x = \frac{π}{2}, then the value of$ $the definite integral \int_{0}^{\infty} f (x - \frac{1}{x}) d x is equal to$