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