Question

If f(x) = `(2 - xcosx)/(2 + xcosx)` and g(x) = log_ex, (x > 0) then the value of the integral `int_((-π)/4)^(π/4) "g"("f"(x))"d"x` is ______.

विकल्प

• log_e1

• log_e2

• log_ee

• log_e3

Answer 1

If f(x) = `(2 - xcosx)/(2 + xcosx)` and g(x) = log_ex, (x > 0) then the value of the integral `int_((-π)/4)^(π/4) "g"("f"(x))"d"x` is `underlinebb(log_e1)`.

Explanation:

f(x) = `(2 - xcosx)/(2 + xcosx)`       `{:("g"(x) = log_"e"x),("h"(x > 0)):}`

`int_((-π)/4)^(π/4) "g"("f"(x))"d"x`

`int_((-π)/4)^(π/4) log_"e"((2 - xcosx)/(2 + xcosx))"d"x`

Let h(x) = `log_"e"  (2 - xcosx)/(2 + x cosx)`

h(–x) = `log_"e"  (2 + xcosx)/(2 - x cosx)`

= `log_"e"((2 - xcosx)/(2 + xcosx))^-1`

= `-log_"e"  (2 - xcosx)/(2 + xcosx)`  = –f(x)

Odd function

For odd function

`int_((-π)/4)^(π/4)"h"(x)` = 0 = log_e1