Evaluate: `int(dx)/(x^(2/3)(1+x^(2/3)))`

Explanation

To evaluate the integral

\[ \int \frac{dx}{x^{2/3}(1+x^{2/3})}, \]

we can follow these steps:

### Step 1: Substitution
Let \( x = t^3 \). Then, the differential \( dx \) can be expressed as:

\[ dx = 3t^2 dt. \]

### Step 2: Rewrite the Integral
Substituting \( x = t^3 \) into the integral, we have:

\[ x^{2/3} = (t^3)^{2/3} = t^2. \]

Thus, the integral becomes:

\[ \int \frac{3t^2 dt}{t^2(1 + t^2)}. \]

### Step 3: Simplify the Integral
The \( t^2 \) in the numerator and denominator cancels out:

\[ \int \frac{3 dt}{1 + t^2}. \]

### Step 4: Integrate
The integral \( \int \frac{dt}{1 + t^2} \) is a standard integral that equals \( \tan^{-1}(t) \). Therefore, we have:

\[ 3 \int \frac{dt}{1 + t^2} = 3 \tan^{-1}(t) + C, \]

where \( C \) is the constant of integration.

### Step 5: Back Substitute
Now, we substitute back \( t = x^{1/3} \):

\[ 3 \tan^{-1}(t) = 3 \tan^{-1}(x^{1/3}) + C. \]

### Final Answer
Thus, the final result of the integral is:

\[ \int \frac{dx}{x^{2/3}(1+x^{2/3})} = 3 \tan^{-1}(x^{1/3}) + C. \]

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