∫ e^x [log(sec x + tan x) + sec x] dx

To solve the integral ∫ e x [ lo g ( sec x + tan x ) + sec x ] d x , we can break it down into steps by evaluating the integral of each component separately.
First, let's recall some important trigonometric identities:

sec x = c o s x 1 ​
tan x = c o s x s i n x ​
sec x + tan x = c o s x 1 + s i n x ​

The structure of the integrand suggests that integration by parts might be a good approach. Integration by parts follows the formula:
∫ u d v = uv − ∫ v d u
Let's set:

u = lo g ( sec x + tan x ) + sec x
d v = e x d x

Then, differentiate u and integrate d v :

d u = ( d x d ​ [ lo g ( sec x + tan x )] + d x d ​ [ sec x ] ) d x
v = e x

Now, let's apply integration by parts:
∫ e x [ lo g ( sec x + tan x ) + sec x ] d x = e x [ lo g ( sec x + tan x ) + sec x ] − ∫ e x ( d x d ​ [ lo g ( sec x + tan x )] + d x d ​ [ sec x ] ) d x
This can be quite complex to evaluate step-by-step without further simplification or substitution. In practice, evaluation of such integrals often requires specific techniques or numerical methods which might be beyond direct integration techniques.
For learning purposes, note that tackling such complex integrals usually involves understanding the behavior of each component—for example, using numerical methods when analytical solutions are difficult or framing the problem differently for more tractable components. Each integral involving trigonometric and exponential functions can pose unique challenges dependent on the complexity of the functions involved.
If you're working through this in a college course, reviewing integration techniques and exploring software tools like computer algebra systems can be very helpful to explore such complex expressions.

Answered by BenjaminOwenLewis | 2025-07-06

To solve the integral ∫ e x [ lo g ( sec x + tan x ) + sec x ] d x , we can use integration by parts, selecting appropriate u and d v terms. After applying the integration by parts formula, we simplify to find the integral of the remaining complex expression. For challenging integrals, utilizing numerical methods or software tools is often beneficial.
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Answered by BenjaminOwenLewis | 2025-07-27