Question

Evaluate the integral by making the given substitution.

\(\int {\frac{{{\rm{se}}{{\rm{c}}^{\rm{2}}}{\rm{(1/x)}}}}{{{{\rm{x}}^{\rm{2}}}}}} {\rm{dx,}}\;\;\;{\rm{u = 1/x}}\).

Short answer

The solution of given integral\(\int {\frac{{{{\sec }^2}(1/x)}}{{{x^2}}}} dx\) is \( - \tan (\frac{1}{x}) + C\).

Step-by-step solution

Substitution method of evaluating integral

Substitution method is a method which transforms given integral into a simple form of integral by substituting the independent variable by others.

Applying substitution method 

Let \(u = \frac{1}{x}\)

\(\begin{aligned}{c}du &= - \frac{1}{{{x^2}}}dx\\ - du &= \frac{{dx}}{{{x^2}}}\end{aligned}\)

Substituting u in given function to get required results-

\(\begin{aligned}{c}\int {\frac{{{{\sec }^2}(1/x)}}{{{x^2}}}} dx &= - \int {({{\sec }^2}u)du} \\ &= - \tan u + C\end{aligned}\)

Replace value of u with original value

\(\)\(\int {\frac{{{{\sec }^2}(1/x)}}{{{x^2}}}} dx = - \tan (\frac{1}{x}) + C\)

Therefore, the solution of given integral\(\int {\frac{{{{\sec }^2}(1/x)}}{{{x^2}}}} dx\) is \( - \tan (\frac{1}{x}) + C\).