Question

Evaluate question 4 only​

Answer 1

Substitute
`y = \sqrt x`, so that
`y^2 = x` and
`2y\,dy = dx`. Then the integral becomes

`\displaystyle \int (dx)/(√(1 + \sqrt x)) = 2 \int \frac y{√(1+y)} \, dy`

Now substitute
`z=1+y`, so
`dz=dy`. The integral transforms to

`\displaystyle 2 \int \frac y{√(1+y)} \, dy = 2 \int (z-1)/(\sqrt z) \, dz = 2 \int \left(\sqrt z - \frac1{\sqrt z}\right) \, dz`

The rest is trivial. By the power rule,

`\displaystyle \int \left(\sqrt z - \frac1{\sqrt z}\right) \, dz = \frac23 z^(3/2) - 2z^(1/2) + C = \frac23 \sqrt z (z - 3) + C`

Put everything back in terms of
`y`, then
`x` :

`\displaystyle 2 \int \frac y{√(1+y)} \, dy = \frac43 √(1+y) (y - 2) + C`

`\displaystyle \int (dx)/(√(1+\sqrt x)) = \boxed{\frac43 √(1+\sqrt x) (\sqrt x - 2) + C}`