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Question
If `int_0^1 "e"^"t"/(1 + "t") "dt"` = a, then `int_0^1 "e"^"t"/(1 + "t")^2 "dt"` is equal to ______.
Options
`"a" - 1 + "e"/2`
`"a" + 1 - "e"/2`
`"a" - 1 - "e"/2`
`"a" + 1 + "e"/2`
Solution
If `int_0^1 "e"^"t"/(1 + "t") "dt"` = a, then `int_0^1 "e"^"t"/(1 + "t")^2 "dt"` is equal to `"a" + 1 - "e"/2`.
Explanation:
Since I = `int_0^1 "e"^"t"/(1 + "t") "dt"`
= `|1/(1 + "t") "e"^"t"|_0^1 + int_0^1 "e"^"t"/(1 + "t")^2 "dt"` = a ...(Given)
Therefore, `int_0^1 "e"^"t"/(1 + "t")^2 = "a" - "e"/2 + 1`.
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