`int(1)/(x^(2))cos((1)/(x))dx`

Answer

To solve the integral \( \int \frac{1}{x^2} \cos\left(\frac{1}{x}\right) dx \), we can use a substitution method. Here are the steps to solve the integral:

### Step 1: Substitution
Let \( t = \frac{1}{x} \). Then, we differentiate \( t \) with respect to \( x \):
\[ dt = -\frac{1}{x^2} dx \quad \Rightarrow \quad dx = -x^2 dt = -\frac{1}{t^2} dt \]
This substitution will help us simplify the integral.

### Step 2: Rewrite the Integral
Now, we can rewrite the integral in terms of \( t \):
\[ \int \frac{1}{x^2} \cos\left(\frac{1}{x}\right) dx = \int \frac{1}{x^2} \cos(t) \left(-\frac{1}{t^2} dt\right) \]
Since \( \frac{1}{x^2} = t^2 \), we have:
\[ = -\int \cos(t) dt \]

### Step 3: Integrate
Now we can integrate:
\[ -\int \cos(t) dt = -\sin(t) + C \]

### Step 4: Substitute Back
Now we substitute back \( t = \frac{1}{x} \):
\[ -\sin\left(\frac{1}{x}\right) + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int \frac{1}{x^2} \cos\left(\frac{1}{x}\right) dx = -\sin\left(\frac{1}{x}\right) + C \]