What is the indefinite integral of dx/[(3radical(x)+4radical(x)]

Question

In the denominator it is a x raised to 1/3 + x raised to the 1/4. I am getting a really long answer and am just trying to make sure I am on the right path?

Answer 1

The key is a substitution like u = x^1/12 with du = (1/12)x^-11/12 dx or 12u¹¹du = dx

The integral becomes ∫12u¹²du/(u⁴ + u³) = 12∫u⁸du/(u+1) after dividing out factor of u³

Do division until you get a remainder of 0th order

12∫(u⁷ - u⁶ +...+ u - 1 + 1/(1+u))du (note alternating sign. All of the parts of the integral are integrable. The last term is ln(1+u) and the rest are power laws. Substitute the x back in and you're all set, and, yes, it has 9 terms in the solution + arbitrary constant.

Good luck!