Question

Evaluate: `int_1^3 sqrt(x + 5)/(sqrt(x + 5) + sqrt(9 - x))dx`

Answer 1

I = `int_1^3 sqrt(x + 5)/(sqrt(x + 5) + sqrt(9 - x))dx`  ...(1)

Replace, f(x) by f(a + b – x) and x by (4 – x)

I = `int_1^3 sqrt(4 - x + 5)/(sqrt(9 - x) + sqrt(x + 5))dx`

= `int_1^3 sqrt(9 - x)/(sqrt(9 - x) + sqrt(x + 5))dx`  ...(2)

Adding equation (i) and (ii)

2I = `int_1^3 sqrt(x + 5)/(sqrt(x + 5) + sqrt(9 - x))dx+int_1^3 sqrt(9 - x)/(sqrt(9 - x) + sqrt(x + 5))dx`

= `int_1^3(sqrt(x+5)+sqrt(9-x))/(sqrt(x+5)+sqrt(9-x)).dx`

= `int_1^31dx=[x]_1^3`

2I = 3 - 1

2I = 2 

I = 1