Use U-substitution to evaluate?

Question

Use U-substitution to evaluate

1/(3(x)^1/2) [6 +(x)^1/2]^1/5 dx

Please show your work

Answer 1

I assume you want an integral...

Substitute u = sqrt x, then dx = d(u^2) = 2u du.

The problem becomes

INT [(6+u)^(1/5) / 3u] 2u du =

= 2/3 INT (6+u)^(1/5) du

= 2/3 . 5/6 . (6+u)^(6/5) + C

= 5/9 . (6+u)^(6/5) + C

= 5/9 . [6 + x^(1/2)]^(6/5)