Question

Integrate the function: 

`1/(x^(1/2) + x^(1/3))  ["Hint:" 1/(x^(1/2) + x^(1/3)) = 1/(x^(1/3)(1+x^(1/6))),  "put x" = t^6]`

Answer 1

Let I = `int 1/(x^(1/2) + x^(1/3))`dx

Put `x^(1/6) = t, x = t^6 `

dx = 6 t⁵ dt

`int = (6 t^5 dt)/(t^3 + t^2) = 6 int (t^3/(t + 1))`dt

`= 6 int (t^3 + 1 - 1)/(t + 1)`dt

`= 6 int ((t^3 + 1)/(t + 1) - 1/(t + 1))  dt`

`= int (((t + 1)(t^2 - t - 1))/(t + 1) - 1/(t + 1)) dt`

`= 6 int (t^2 - t + 1 - 1/(t + 1))  dt`

`= 6 int t^2 dt - 6 int  t dt - 6 int 1/(t + 1)  dt`

`= (6t^3)/3 - (6t^2)/2 - 6 log |t + 1| + C`

= 2t³ - 3t² - 6 log |t + 1| + C

= `2sqrtx - 3x^(1/3) + 6x^(1/6) - 6 log |x^(1/6) + 1| + C`