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Question
Evaluate: `int_0^9 sqrt(x)/(sqrt(x) + sqrt(9 - x) "d"x`
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Solution
Let I = `int_0^9 sqrt(x)/(sqrt(x) + sqrt(9) - x) "d"x` ........(i)
= `int_0^9 (sqrt(9 - x))/(sqrt(9 - x) + sqrt(9 - (9 -x))) "d"x` ........`[∵ int_0^"a" "f"(x)"d"x = int_0^"a" "f"("a" - x)"d"x]`
∴ I = `int_0^9 (sqrt(9 - x))/(sqrt(9 - x) + sqrt(x)) "d"x` ......(ii)
Adding (i) and (ii), we get
2I = `int_0^9 (sqrt(x))/(sqrt(x) + sqrt(9 - x)) "d"x + int_0^9 (sqrt(9 - x))/(sqrt(9 - x) + sqrt(x)) "d"x`
= `int_0^9 (sqrt(x) + sqrt(9 - x))/(sqrt(x) + sqrt(9 - x)) "d"x`
= `int_0^9 "d"x`
= `[x]_0^9`
∴ 2I = 9 − 0
∴ I = `9/2`
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