Question

How do you integrate `int 1/(x^(3/2) + x^(1/2)) dx`?

Answer 1

`2arc tansqrtx+C.`

Explanation:

Let `I=int1/(x^(3/2)+x^(1/2))dx=int1/{sqrtx(x+1)}dx`.

We take subst. `sqrtx=t, or, x=t^2 rArr dx=2tdt.`

`:. I=int1/(t(t^2+1))2tdt=2int1/(t^2+1)dt=2arc tant`

Hence, `I=2arc tansqrtx+C,` as Respected Eric Sia has derived!