Evaluate: ∫x⋅e2x dx - Mathematics and Statistics

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Sum

Evaluate: `int "x" * "e"^"2x"` dx

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Solution

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

`= 1/2 "xe"^"2x" - 1/2 int "e"^"2x"` dx

`= 1/2 "x e"^"2x" - 1/2 * "e"^"2x"/2` + c

∴ I = `1/4 "e"^"2x" ("2x" - 1)` + c

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Chapter 5: Integration - Miscellaneous Exercise 5 [Page 139]

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Balbharati Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board
Chapter 5 Integration
Miscellaneous Exercise 5 | Q 4.4 | Page 139
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