Question

Evaluate the following integrals : `int_2^7 sqrt(x)/(sqrt(x) + sqrt(9 - x))*dx`

Evaluate: `∫_2^7 sqrtx/(sqrtx + sqrt(9 − x))dx`

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            ...[∵ 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)`