Question

If `int(2x^(1/2))/(x^2)  dx = k  .  2^(1/x) + C`, then k is equal to ______.

Options

• `(-1)/(log 2)`

• − log 2

• −1

• `1/2`

Answer 1

If `int(2x^(1/2))/(x^2) dx = k  .  2^(1/x) + C`, then k is equal to `bbunderline((-1)/(log 2))`.

Explanation:

`int(2x^(1/2))/(x^2)  dx = k  .  2^(1/x) + C    ...(1)`

Put `1/x = t`

`(-1)/x  dx = dt`

`dx = -x^2  dt`

= `int(2^t)/(x^2) (-x^2) dt`

= `-int 2^t dt`

= `(-2^t)/(log 2) + C`

Given, `int(2x^(1/2))/(x^2) dx = k  .  2^(1/x) + C`

`(-2^(1/x))/(log 2) + C = K  .  2^(1/x) + C`

Hence, K = `(-1)/(log 2)`