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प्रश्न
Evaluate the definite integral:
`int_0^1 dx/(sqrt(1+x) - sqrtx)`
उत्तर
Let I = `int_0^1 dx/(sqrt(1 + x) - sqrtx)`
On multiplying the numerator and denominator by `sqrt(1 + x) - sqrtx`
I = int_0^1 (sqrt(1 + x) - sqrtx)/(1 + x - x) dx`
`= int_0^1 (sqrt(1 + x) - sqrtx) dx`
`= int_0^1 sqrt(1 + x) dx + int_0^1 sqrtx dx`
`= [2/3 (1 + x)^(3//2)]_0^1 + [2/3 x^(3//2)]_0^1`
`= 2/3 (2^(3//2) - 1) + 2/3 [1 - 0]`
`= 2/3 * 2^(3//2) - 2/3 + 2/3`
`= 2/3 * 2^(3//2)`
`= 2/3 * 2sqrt2`
`= (4sqrt2)/3`
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