6. $int\frac{sec^{2}(1/x)}{x^{2}}dx$, $u = 1/x$

Explanation:

Step1: Find the derivative of $u$

Given $u = \frac{1}{x}$, then $du=-\frac{1}{x^{2}}dx$.

Step2: Substitute $u$ and $du$ into the integral

The integral $\int\frac{\sec^{2}(\frac{1}{x})}{x^{2}}dx$ becomes $-\int\sec^{2}(u)du$.

Step3: Integrate $\sec^{2}(u)$

We know that the antiderivative of $\sec^{2}(u)$ is $\tan(u)+C$. So $-\int\sec^{2}(u)du=-\tan(u)+C$.

Step4: Substitute back $u = \frac{1}{x}$

The result is $-\tan(\frac{1}{x})+C$.

Answer:

$-\tan(\frac{1}{x})+C$