Evaluate the following integrals.
`int (dx)/(x cos^(2) (log x))`

Answer

To evaluate the integral

\[ I = \int \frac{dx}{x \cos^2(\log x)}, \]

we will use the substitution method.

### Step 1: Substitution
Let

\[ t = \log x. \]

Then, differentiating both sides gives us:

\[ \frac{1}{x} dx = dt \quad \Rightarrow \quad dx = x \, dt = e^t \, dt. \]

### Step 2: Rewrite the Integral
Now, substituting \(x = e^t\) into the integral, we have:

\[ I = \int \frac{e^t \, dt}{e^t \cos^2(t)} = \int \frac{dt}{\cos^2(t)}. \]

### Step 3: Simplify the Integral
The integral simplifies to:

\[ I = \int \sec^2(t) \, dt. \]

### Step 4: Integrate
The integral of \(\sec^2(t)\) is:

\[ I = \tan(t) + C, \]

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

### Step 5: Back Substitute
Now, we substitute back \(t = \log x\):

\[ I = \tan(\log x) + C. \]

### Final Answer
Thus, the value of the integral

\[ \int \frac{dx}{x \cos^2(\log x)} = \tan(\log x) + C. \]

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