Evaluate the following integrals :
(a) `int ( root( 3) (1+ "In" x) )/( x) dx`,

Answer

To evaluate the integral

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

we will use substitution and integration techniques. Here is the step-by-step solution:

### Step 1: Substitution
Let

\[ t = 1 + \ln x. \]

Then, differentiate both sides with respect to \(x\):

\[ \frac{dt}{dx} = \frac{1}{x} \implies dt = \frac{1}{x} \, dx. \]

This means we can rewrite \(dx\) in terms of \(dt\):

\[ dx = x \, dt = e^{t-1} \, dt \quad \text{(since } x = e^{t-1} \text{ from } t = 1 + \ln x \text{)}. \]

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

\[ \int \frac{(1 + \ln x)^{\frac{1}{3}}}{x} \, dx = \int t^{\frac{1}{3}} \, dt. \]

### Step 3: Integrate
Now we can integrate \(t^{\frac{1}{3}}\):

\[ \int t^{\frac{1}{3}} \, dt = \frac{t^{\frac{1}{3} + 1}}{\frac{1}{3} + 1} + C = \frac{t^{\frac{4}{3}}}{\frac{4}{3}} + C = \frac{3}{4} t^{\frac{4}{3}} + C. \]

### Step 4: Substitute Back
Now we substitute back \(t = 1 + \ln x\):

\[ \frac{3}{4} (1 + \ln x)^{\frac{4}{3}} + C. \]

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

\[ \int \frac{(1 + \ln x)^{\frac{1}{3}}}{x} \, dx = \frac{3}{4} (1 + \ln x)^{\frac{4}{3}} + C. \]

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