Find the integral of the following function
$\int{\sec x\left( \sec x+\tan x \right)dx}$.

Hint: Use the basic integration $\int{{{\sec }^{2}}x.dx=\tan x+c}$ and $\int{\sec x\tan xdx=\sec x+c}$ to solve this question. Solve it as two separate integrals to make it easier.

Complete step-by-step answer:
It is given in the question that we have to integrate $\int{\sec x\left( \sec x+\tan x \right)dx}$.

To start integrating the given function, we have to name the integral. Therefore, we consider it as $I=\int{\sec x\left( \sec x+\tan x \right)dx}$ .

Now we can expand the terms by taking the term sec x inside the bracket. Then, we will get;
$I=\int{\left( {{\sec }^{2}}x+\sec x.\tan x \right)dx}$

Since we have the sum of two functions inside the integral, we can split them as two different integrals. Therefore, the above integral can also be written as,
 $I=\int{\left( {{\sec }^{2}}x \right)dx+\int{\left( \sec x.\tan x \right)}}dx$

Looking at the terms inside the integral, we can clearly notice that these are in the form of standard formulas for integration of functions. Now, we can apply the basic integration formulas for these integrals. We know that the basic formulas are given by,
$\int{{{\sec }^{2}}x.dx=\tan x+c}$ and $\int{\sec x\tan x=\sec x+c}$
So, from this we can write the integrals as,
$\begin{align}
  & I=\int{\left( {{\sec }^{2}}x \right)dx}+\int{\left( \sec x.\tan x \right)}dx \\
 & I=\tan x+\sec x+c \\
\end{align}$

Since this is indefinite integration, a constant term denoted by C will also be present in the result.

Therefore, we have found out the Integral of $\sec x\left( \sec x+\tan x \right)dx=\tan x+\sec x+c$.

Note: This question can be solved in just a few steps by using basic integration rules and formulas. So, the student must make sure to memorize all the basic integration formulae. This will save the time for solving the question. Also, integration is the reverse of differentiation. So, the student can even check if the obtained answer is correct by differentiating the obtained result with respect to x. If the student gets the question as the answer after differentiation, the solution would also be correct.