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Question
Evaluate:
`int_2^8 (sqrt(10 - "x"))/(sqrt"x" + sqrt(10 - "x")) "dx"`
Solution
Given, `int_2^8 (sqrt(10 - "x"))/(sqrt"x" + sqrt(10 - "x")) "dx"`
Let I = `int_2^8 (sqrt(10 - "x"))/(sqrt"x" + sqrt(10 - "x")) "dx"` ............(i)
Then using property:
`int_"a"^"b" "f"("x") "dx" = int_"a"^"b" "f"("a" + "b" - "x") "dx"`
I = `int_2^8 (sqrt(10 - (2 + 8 - "x")))/(sqrt(2 + 8 - "x") + sqrt(10 - (2 + 8 - "x"))) "dx"`
= `int_2^8 (sqrt"x")/(sqrt(10 - "x") + sqrt"x") "dx"` ...........(ii)
Adding equation (i) and (ii), we get
2I = `int_2^8 (sqrt(10 - "x") + sqrt"x")/(sqrt"x" + sqrt(10 - "x")) "dx"`
⇒ 2I = `int_2^8 1. "dx"`
⇒ 2I = `["x"]_2^8`
⇒ 2I = 8 − 2
⇒ 2I = 6
∴ I = 3
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