Integrate:– \Huge{{\large{∫} \frac{dx}{ \sqrt{x √(x) - x {}^(2) } }}} ​

Question

Integrate:–

`\Huge{{\large{∫} \frac{dx}{ \sqrt{x √(x) - x {}^(2) } }}}`
​

Answer 1

9514 1404 393

Answer:

4arcsin(x^(1/4)) +C

Explanation:

Let x = z^4. Then dx = 4z^3·dz, and the given expression is ...

`\displaystyle\frac{dx}{\sqrt{x√(x)-x^2}}=(4z^3\,dz)/(√(z^4z^2-z^8))=4\cdot(dz)/(√(1-z^2))`

and its integral is ...

`\displaystyle\int{(4\,dz)/(√(1-z^2))}=4\arcsin{(z)}+C=\boxed{4\arcsin{\sqrt[4]{x}}+C}`

__

Note that the original integrand is only defined on the interval (0, 1).