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Question
Evaluate the following.
`int (1 + "x")/("x" + "e"^"-x")` dx
Solution
Let I = `int (1 + "x")/("x" + "e"^"-x")` dx
`= int (1 + "x")/("x" + 1/"e"^"x")` dx
`= int (1 + "x")/(("x" * "e"^"x" + 1)/"e"^"x")`dx
`= int ("e"^"x"(1 + "x"))/("x" * "e"^"x" + 1)` dx
Put `"x" * "e"^"x" + 1 = "t"`
∴ `["x" * ("e"^"x") + "e"^"x" (1) + 0]`dx = dt
∴ `"e"^"x" ("x" + 1)`dx = dt
∴ I = `int "dt"/"t"`
= log |t| + c
∴ I = log `|"x" * "e"^"x" + 1|` + c
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