Question

Choose the correct alternative :

`int_2^7 sqrt(x)/(sqrt(x) + sqrt(9 - x))*dx` =

Options

• `(7)/(2)`

• `(5)/(2)`

• 7

• 2

Answer 1

Let I = `int_2^7 sqrt(x)/(sqrt(x) + sqrt(9 - x))*dx`         ...(i)

= `int_2^7 sqrt(2 + 7 - x)/(sqrt(2 + 7 - x) + sqrt(9 - (2 + 7 - x)))*dx        ...[because int_"a"^"b" f(x)*dx = int_"a"^"b" f("a" + "b" - x)*dx]`

∴ I = `int_2^7 sqrt(9 - x)/(sqrt(9 - x) + sqrt(x))*dx`      ...(ii)

Adding (i) and (ii), we get

2I = `int_2^7 sqrt(x)/(sqrt(x) + sqrt(9 - x))*dx + int_2^7 sqrt(9 - x)/(sqrt(9 - x) + sqrt(x))*dx` 

= `int_2^7 (sqrt(x) + sqrt(9 - x))/(sqrt(x) + sqrt(9 - x))*dx`

= `int_2^7 1*dx`

= `[x]_2^7`
∴ 2I = 7 – 2 = 5
∴ I = `(5)/(2)`.