State whether the following statement is True or False. If ∫x e2x dx is equal to e2x f(x) + c, where c is constant of integration, then f(x) is 2x-12. - Mathematics and Statistics

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MCQ
True or False

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

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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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Chapter 1.5: Integration - Q.3

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