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Question
Evaluate the following integrals using properties of integration:
`int_(-5)^5 x cos(("e"^x - 1)/("e"^x + 1)) "d"x`
Solution
Let f(x) = `x cos (("e"^x - 1)/("e"^x + 1)) "d"x`
f(−x) = `(- x)cos (("e"^-x - 1)/("e"^-x + 1))`
= `(- x)cos ((1/"e"^x - 1)/(1/"e"^x + 1))`
= `(- x)cos ((1 - "e"^x)/(1 + "e"^x))`
= `(- x)cos ((-("e"^x - 1))/("e"^x + 1))`
= `- x cos(("e"^x - 1)/("e"^x + 1))`
f(x) = f(−x)
f(x) is an odd function
`int_(-"a")^"a" "f"(x) "d"x` = 0
∴ `int_(-5)^5 x cos (("e"^x - 1)/("e"^x + 1)) "d"x` = 0
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