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प्रश्न
Given `int "e"^"x" (("x" - 1)/("x"^2)) "dx" = "e"^"x" "f"("x") + "c"`. Then f(x) satisfying the equation is:
विकल्प
x
x2
`1/"x"`
None of the above options
उत्तर
`1/"x"`
Explanation:
Given, `int "e"^"x" (("x" - 1)/("x"^2)) "dx" = "e"^"x" "f"("x") + "c"`
Taking L.H.S. = `int "e"^"x" (("x" - 1)/"x"^2) "dx"`
= `int "e"^"x" (1/"x" - 1/"x"^2) "dx"`
= `int "e"^"x". 1/"x" "dx" - int "e"^"x". 1/"x"^2 dx"`
Integrating the first integral by parts taking `1/"x"` as the first function,
= `1/"x". "e"^"x" + int 1/"x"^2. "e"^"x" "dx" - int "e"^"x". 1/"x"^2 "dx" + "c"`
= `1/"x". "e"^"x" + "c"`
On comparing with the R.H.S., we get
f(x) = `1/"x"`
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