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Question
Integrate the following with respect to x.
`(x^("e" - 1) + "e"^(x - 1))/(x^"e" + "e"^x)`
Solution
`(x^("e" - 1) + "e"^(x - 1))/(x^"e" + "e"^x) = (x^("" 1) + "e"^x/"e")/(x^"e" + "e"^x)`
= `("e"x^("e" - 1) + "e"^x)/("e"(x^"e" + "e"^x))`
Let f(x) = xe + ex
Then f'(x) = `"e"x^("e" - 1) + "e"^x`
So `int (x^("e" - 1) + "e"^(x - 1))/(x^"e" + "e"^x) "d"x = int ("e"x^("e" - 1) + "e"^x)/("e"(x^"e" + "e"^x)) "d"x`
= `1/"e" int ("f'"(x))/("f"(x)) "d"x`
= `1/"e" log |"f"(x)| + "c"`
= `1/"e" log|x^"e" + "e"^x| + "c"`
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