Evaluate the following integral. etan⁡xcos2xdx

Step 1
Given integral is
${\displaystyle \frac{e^{\tan x}}{{\cos }^{2}x}dx}$
We know that ${\displaystyle \sec x=\frac{1}{\cos x}}$
Therefore, given integral can be written as
${\displaystyle \int \frac{e^{\tan x}}{{\cos }^{2}x}dx=\int e^{\tan x}\left({\sec }^{2}xdx\right)}$
Step 2
Now, we will use the following substitution to evaluate the integral
$\tan x=t$
${\displaystyle {\sec }^{2}xdx=dt}$
Therefore, we have
${\displaystyle \int \frac{e^{\tan x}}{{\cos }^{2}x}dx=\int e^{t}dt}$
${\displaystyle \int \frac{e^{\tan x}}{{\cos }^{2}x}dx=e^t+c}$
${\displaystyle \int \frac{e^{\tan x}}{{\cos }^{2}x}dx=e^{\tan x}+c}$
Step 3
Ans:
${\displaystyle \int \frac{e^{\tan x}}{{\cos }^{2}x}dx=e^{\tan x}+c}$

Given:
${\displaystyle \int \frac{e^{\tan \left(x\right)}}{{\cos \left(x\right)}^2}dx}$
${\displaystyle \int 1dt}$
Use ${\displaystyle \int 1dx=x}$ to evaluate the integral
t
Substitute back
${\displaystyle e^{\tan \left(x\right)}}$
Add C
Answer:
${\displaystyle e^{\tan \left(x\right)}+C}$

$\int \frac{e^{\tan (x)}}{\cos (x{)}^2}dx$
We put the expression $1/\cos (x{)}^2$ under the differential sign, i.e.:
$\frac{1}{\cos (x{)}^2}dx=d(\tan (x)),t=\tan (x)$
Then the original integral can be written as follows:
$\int e^{t}dt$
This is a tabular integral:
$\int e^{t}dt=e^t+C$
To write down the final answer, it remains to substitute $\tan (x)$ instead of t.
$e^{\tan (x)}+C$