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Question
Evaluate the following integrals : `int_2^5 sqrt(x)/(sqrt(x) + sqrt(7 - x))*dx`
Solution
Let I = `int_2^5 sqrt(x)/(sqrt(x) + sqrt(7 - x))*dx` ...(i)
= `int_2^5 sqrt(2 + 5 - x)/(sqrt(2 + 5 - x) + sqrt(7 - (2 + 5 - x)))*dx ...[because int_"a"^"b" f(x)*dx = int_"a"^"b" f("a" + "b" - x)*dx]`
∴ I = `int_2^5 sqrt(7 - x)/(sqrt(7 - x) + sqrt(x))*dx` ...(ii)
Adding (i) and (ii), we get
2I = `int_2^5 sqrt(x)/(sqrt(x) + sqrt(7 - x))*dx + int_2^5 sqrt(7 - x)/(sqrt(7 - x) + sqrt(x))*dx`
= `int_2^5 (sqrt(x) + sqrt(7 - x))/(sqrt(x) + sqrt(7 - x))*dx`
= `int_2^5 1*dx`
= `[x]_2^5`
∴ 2I = 5 – 2 = 3
∴ I = `(3)/(2)`.
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