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प्रश्न
Evaluate the following.
`int "x"^3 "e"^("x"^2)`dx
उत्तर
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
The answer in the textbook is incorrect.
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