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Question
State whether the following statement is True or False.
If `int x "e"^(2x)` dx is equal to `"e"^(2x)` f(x) + c, where c is constant of integration, then f(x) is `(2x - 1)/2`.
Options
True
False
Solution
False
Explanation:
Let I = `int "x" * "e"^"2x"` dx
`= "x" int "e"^"2x" * "dx" - int ["d"/"dx" ("x") int "e"^"2x" * "dx"]` dx
`= "x" * "e"^"2x"/2 - int 1 * "e"^"2x"/2 * "dx"`
`= "x"/2 "e"^"2x" - 1/2 int "e"^"2x" +` c
`= "x"/2 "e"^"2x" - 1/2 * "e"^"2x"/2` + c
`= "e"^"2x" ("x"/2 - 1/4)` + c
`= "e"^"2x" (("2x" - 1)/4)` + c
∴ f(x) = `(2"x" - 1)/4`
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