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प्रश्न
If `int_0^1 ("d"x)/(sqrt(1 + x) - sqrt(x)) = "k"/3`, then k is equal to ______.
विकल्प
`sqrt(2)(2sqrt(2) - 2)`
`sqrt(2)/3(2 - 2sqrt(2))`
`(2sqrt(2) - 2)/3`
`4sqrt(2)`
उत्तर
If `int_0^1 ("d"x)/(sqrt(1 + x) - sqrt(x)) = "k"/3`, then k is equal to `bb(underline(4sqrt(2))`.
Explanation:
Step 1: Rationalizing the Denominator
`sqrt(1+x)-sqrtx = ((1+x)-x)/(sqrt(1+x) + sqrtx) = 1/(sqrt(1+x) + sqrtx)`
`I = int_0^1(sqrt(1+x) + sqrtx)dx`
Step 2: Evaluating Each Integral
`I = int_0^1 sqrt(1+x) dx + int_0^1sqrtx dx`
Using standard integration formulas:
`(4sqrt2 - 2)/3`
`2/3`
`I = (4sqrt2 - 2)/3 + 2/3 = (4sqrt2)/3`
Step 3: Finding k
`I = k/3`
`k = 4sqrt2`
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