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Question
Evaluate: `int_1^3 sqrt(x)/(sqrt(x) + sqrt(4) - x) dx`
Solution
Let I = `int_1^3 sqrt(x)/(sqrt(x) + sqrt(4) - x)` ...(i)
Using property `int_a^b f(x)dx = int_a^b f(a + b - x)dx`, we get
I = `int_1^3 sqrt(4 - x)/(sqrt(4 - x) + sqrt(x))dx` ...(ii)
On adding equations (i) and (ii}, we get
2I = `int_1^3 (sqrt(x) + sqrt(4 - x))/(sqrt(x) + sqrt(4 - x))dx`
= `int_1^3 1dx`
= `[x]_1^3`
= 3 – 1 = 2
∴ I = 1
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