What is the value of `int(sqrtx + x)^(-1) dx`?

Options

A. `ln (x + sqrtx) + c`
B. `2 ln (1 + sqrtx) + c`
C. `2 ln (x + sqrtx) + c`
D. `2 ln (x - sqrtx) + c`

Answer

B

Explanation

`int (x + sqrtx)^(-1) dx = int (1)/((x + sqrtx)) dx`
`= int (1)/((sqrtx.sqrtx + sqrtx)).dx`
Let `sqrtx + 1 = t`
then, `(1)/(2sqrtx) dx = dt rArr (1)/(sqrtx) dx = 2dt`
`:. Int (1)/(sqrtx (sqrtx + 1))dx = int (2dt)/(t) = 2 log t + c`
`= 2 log (1 + sqrtx) + c`