Question

Evaluate the following integrals : `int_2^5 sqrt(x)/(sqrt(x) + sqrt(7 - x))*dx`

Answer 1

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)`.