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

Answer

To evaluate the integral

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

we will follow these steps:

### Step 1: Substitution
Let \( t = \log x \). Then, the differential \( dx \) can be expressed in terms of \( dt \) as follows:

\[ dx = e^t \, dt. \]

Since \( x = e^t \), we also have

\[ \frac{1}{x} = \frac{1}{e^t}. \]

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

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

### Step 3: Simplify the Integral
The integral can be simplified using the identity \( \frac{1}{\cos^2(t)} = \sec^2(t) \):

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

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

\[ \int \sec^2(t) \, dt = \tan(t) + C, \]

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

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

\[ \tan(t) = \tan(\log x). \]

Thus, the final result of the integral is:

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

### Final Answer
The evaluated integral is:

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

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