Question

If `int_0^1 ("d"x)/(sqrt(1 + x) - sqrt(x)) = "k"/3`, then k is equal to ______.

Options

• `sqrt(2)(2sqrt(2) - 2)`

• `sqrt(2)/3(2 - 2sqrt(2))`

• `(2sqrt(2) - 2)/3`

• `4sqrt(2)`

Answer 1

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`