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प्रश्न
`int_"a"^"b" sqrtx/(sqrtx + sqrt("a" + "b" - x)) "dx"` = ______.
पर्याय
a + b
`("b - a")/2`
a - b
`("a - b")/2`
MCQ
उत्तर
`int_"a"^"b" sqrtx/(sqrtx + sqrt("a" + "b" - x)) "dx"` = `underline(("b - a")/2)`.
Explanation:
Let I = `int_"a"^"b" sqrtx/(sqrtx + sqrt("a" + "b" - x)) "dx"` ...(i)
`= int_"a"^"b" (sqrt("a + b - x"))/(sqrt("a + b - x") + sqrt("a + b" - ("a + b - x")))`
I = `int_"a"^"b" (sqrt("a + b - x"))/(sqrt("a + b - x") + sqrtx)` ....(ii)
On adding Eqs. (i) and (ii), we get
2I = `int_"a"^"b" (sqrtx + sqrt("a + b - x"))/(sqrtx + sqrt("a + b - x"))`dx
`=> "2I" - int_"a"^"b" "dx" = (x)_"a"^"b" = "b - a"`
`=> "I" = ("b - a")/2`
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