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Question
State whether the following statement is True or False.
If ∫ x f(x) dx = `("f"("x"))/2`, then find f(x) = `"e"^("x"^2)`
Options
True
False
Solution
True
Explanation:
If f(x) = `"e"^("x"^2)`, then
`int "x" * "f"("x") "dx" = int "x" * "e"^("x"^2) *` dx
Put x2 = t
∴ 2x dx = dt
∴ x dx = `1/2` dt
∴ `int "x" * "f"("x") "dx" = 1/2 int "e"^"t" * "dt"`
`= 1/2 "e"^"t" + "c"`
`= 1/2 "e"^("x"^2)` + c
`= 1/2` f(x) + c
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