Maharashtra State BoardHSC Commerce 12th Board Exam
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Evaluate the following. ∫x3ex2dx - Mathematics and Statistics

Sum

Evaluate the following.

`int "x"^3 "e"^("x"^2)`dx

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Solution

Let I = `int "x"^3 "e"^("x"^2)`dx

`= int "x"^2 * "x" * "e"^("x"^2)` dx

Put x2 = t

∴ `"2x" * "dx" = "dt"`

∴ x dx = `"dt"/2`

∴ I = `1/2 int "te"^"t"` dt

`= 1/2 ["t" int "e"^"t"  "dt" - int ["d"/"dt" ("t") int "e"^"t" "dt"] "dt"]`

`= 1/2 ["te"^"t" - int 1 * "e"^"t" "dt"]`

`= 1/2 ("te"^"t" - "e"^"t") + "c" = 1/2 "e"^"t" ("t - 1")` + c

∴ I = `1/2  "e"^("x"^2) ("x"^2 - 1)` + c

Notes

[Note: Answer in the textbook is incorrect.]

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