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Question
Evaluate: `int "e"^sqrt"x"` dx
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Solution
Let I = `int "e"^sqrt"x"` dx
Put `sqrt"x"` = t
∴ x = t2
∴ dx = 2t dt
∴ I = `int "e"^"t" * "2t"`dt
`= 2 int "t" * "e"^"t" * "dt"`
`= 2 ["t" int "e"^"t" "dt" - int {"d"/"dx" ("t") int "e"^"t" * "dt"}"dt"]`
`= 2 ["t" * "e"^"t" - int 1 * "e"^"t" "dt"]`
`= 2("te"^"t" - "e"^"t")` + c
`= 2"e"^"t" ("t - 1")` + c
∴ I = `2"e"^sqrt"x" (sqrt"x" - 1)` + c
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