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प्रश्न
Evaluate the following.
`int "x"^2 "e"^"4x"`dx
उत्तर
Let I = `int "x"^2 "e"^"4x"`dx
`= "x"^2 int "e"^"4x" "dx" - int["d"/"dx" ("x"^2) int "e"^"4x" "dx"]` dx
`= "x"^2 * "e"^"4x"/4 - int 2"x" * "e"^"4x"/4` dx
`= ("x"^2 * "e"^"4x")/4 - 1/2 int "x" * "e"^"4x"` dx
`= ("x"^2 * "e"^"4x")/4 - 1/2 ["x" int "e"^"4x" "dx" - int ("d"/"dx" ("x") int "e"^"4x" "dx") "dx"]`
`= ("x"^2 * "e"^"4x")/4 - 1/2 ["x" * "e"^"4x"/4 - int 1 * "e"^"4x"/4 "dx"]`
`= ("x"^2 "e"^"4x")/4 - 1/2[("x" * "e"^"4x")/4 - 1/4 int "e"^"4x" "dx"]`
`= ("x"^2 "e"^"4x")/4 - 1/2[("x" * "e"^"4x")/4 - 1/4 * "e"^"4x"/4]` + c
`= ("x"^2 "e"^"4x")/4 - ("x" "e"^"4x")/8 + "e"^"4x"/32` + c
∴ I = `("e"^"4x")/4 ["x"^2 - "x"/2 + 1/8]` + c
Notes
The answer in the textbook is incorrect.
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