Integrate the functions`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`]

Explanation

To solve the integral \( I = \int \frac{1}{x^{1/2} + x^{1/3}} \, dx \), we will follow the hint provided and perform a substitution.

### Step 1: Factor out \( x^{1/3} \)
We start by rewriting the integrand:
\[ I = \int \frac{1}{x^{1/2} + x^{1/3}} \, dx = \int \frac{1}{x^{1/3}(x^{1/6} + 1)} \, dx \]
This simplifies to:
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