Question

Evaluate: `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`  ...`[∵ 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 equations (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

2I = 3

I = `3/2`