Question

What is the integral of $sec\left(x\right)$?

Answer 1

Find integration of $\sec \left(x\right)$.

Reframe the integral:

$$\begin{gathered} \int \sec xdx=\int \sec x\left(\frac{\sec x+\tan x}{\sec x+\tan x}\right)dx \\ \int \sec xdx=\int \frac{{\sec }^{2}x+\sec x\tan x}{\sec x+\tan x}dx...\left(1\right) \end{gathered}$$

$$\begin{gathered} \mathrm{Take},\sec x+\tan x=t \\ \Rightarrow \left(\sec x\tan x+{\sec }^{2}x\right)dx=dt \end{gathered}$$.

To Find Put the above values in the equation $\left(1\right)$:

$$\begin{gathered} \int \sec xdx=\int \frac{dt}{t} \\ \int \sec xdx=\ln |t|+C\left\{\because \int \frac{dt}{t}=\ln \left|t\right|\right\} \\ \int secxdx=\ln \left|\left(secx+\tan x\right)\right|+C\left\{\mathrm{Put}t=\sec x+\tan x\right\} \end{gathered}$$

Hence, the required integral is $\int secxdx=\ln \left|\left(secx+\tan x\right)\right|+C$ where $C$ is a constant of integration .