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Question
Evaluate: `int_3^8 (11 - x)^2/(x^2 + (11 - x)^2) "d"x`
Solution
Let I = `int_3^8 (11 - x)^2/(x^2 + (11 - x)^2) "d"x` .......(i)
= `int_3^8 ([11 - (1 - x)]^2)/((11 - x)^2 + [11 - (11 - x)]2) "d"x` ........`[∵ int_"a"^"b" "f"(x)"d"x = int_"a"^"b" "f"("a" + "b" - x)"d"x]`
∴ I = `int_3^8 x^2/((11 - x)^2 + x^2) "d"x` .......(ii)
Adding (i) and (ii), we get
2I = `int_3^8 (11 - x)^2/(x^2 + (11 - x)^2) "d"x + int_3^8 x^2/((11 - x)^2 + x^2) "d"x`
= `int_3^8 ((11 - x)^2 + x^2)/(x^2 + (11 - x)^2) "d"x`
∴ 2I = `int_3^8 1. "d"x`
∴ I= `1/2[x]_3^8`
∴ I = `1/2(8 -3)`
∴ I =`5/2`
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