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$\int {{e^x}} \sec x(1 + \tan x)dx$.

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Hint: In the above given question, we have to find out the integral by using the conditions involving both the exponential as well as the trigonometric functions. We must keep in mind the properties of both of these and the formula associated with these like $\int {{e^x}(f(x) + f'(x)} )dx = {e^x}f(x)$ to reach the solution.

We have the given integral as

$\int {{e^x}} \sec x(1 + \tan x)dx$

This can be written as

$ = \int {{e^x}(} \sec x + \sec x\tan x)dx$ … (1)

Now we know that the derivative of\[(\sec x)\] can be given as

\[ \Rightarrow f'(\sec x) = (\sec x)dx = \sec x\tan x\] … (2)

We have the standard formula as

$\int {{e^x}(f(x) + f'(x)} )dx = {e^x}f(x)$.

Where $f(x) = \sec x $and$ f'(\sec x)dx = \sec x\tan x$

Now after comparing equation (1) and (2) with the standard formula, we get the value of the integral as

\[ = {e^x}\sec x\]

Therefore, we have the integral $\int {{e^x}} \sec x(1 + \tan x)dx = {e^x}\sec x$.

Note: Whenever we face such types of problems the key point is to have a good grasp of the integration formula. Also remember that the integration of $ {e^x} $is given as$\int {{e^x}dx = {e^x}} $. With the help of these we can easily reach our solution.

We have the given integral as

$\int {{e^x}} \sec x(1 + \tan x)dx$

This can be written as

$ = \int {{e^x}(} \sec x + \sec x\tan x)dx$ … (1)

Now we know that the derivative of\[(\sec x)\] can be given as

\[ \Rightarrow f'(\sec x) = (\sec x)dx = \sec x\tan x\] … (2)

We have the standard formula as

$\int {{e^x}(f(x) + f'(x)} )dx = {e^x}f(x)$.

Where $f(x) = \sec x $and$ f'(\sec x)dx = \sec x\tan x$

Now after comparing equation (1) and (2) with the standard formula, we get the value of the integral as

\[ = {e^x}\sec x\]

Therefore, we have the integral $\int {{e^x}} \sec x(1 + \tan x)dx = {e^x}\sec x$.

Note: Whenever we face such types of problems the key point is to have a good grasp of the integration formula. Also remember that the integration of $ {e^x} $is given as$\int {{e^x}dx = {e^x}} $. With the help of these we can easily reach our solution.