Question

Question #eab38

Answer 1

`6(x^(7/6)/7-x^(5/6)/5+x^(1/2)/3-x^(1/6)+arctanx^(1/6))+c`

Explanation:

Use `z=x^(1/6)\ \ => x=z^6, sqrt(x)=z^3, root(3)(x)=z^2, dx=6z^5dz`
`intsqrtx/(1+root(3)x)dx=6intz^8/(1+z^2)dz`

Then

`z^8/(1+z^2)=z^6-z^4+z^2-1+1/(a+z^2)`

so

`intsqrtx/(1+root(3)x)dx=6(z^7/7-z^5/5+z^3/3-z+arctan z)+c`

then make substitution `z=x^(1/6)`