∫ 1 / sec x + tan x dx =

• 1 / sec x + c
• sec x + log(sec x + tan x) + c
• cos x + log(cos x − cot x) + c
• log(sec x) + c

The integral you provided, ∫ (1 / (sec x + tan x)) dx, can be simplified using a substitution method. Here’s a breakdown of how to approach it.

Understanding the Integral

The expression sec x + tan x can be rewritten in terms of sine and cosine:

• sec x = 1/cos x
• tan x = sin x/cos x

This means that sec x + tan x = (1 + sin x) / cos x.

Substitution Method

To solve the integral, we can use the substitution:

• Let u = sec x + tan x.
• Then, the derivative du = (sec x tan x + sec^2 x) dx.

This allows us to express dx in terms of du, making the integral easier to handle.

Final Steps

After performing the substitution and simplifying, you will arrive at the integral's solution. The final result will include logarithmic terms and constants, which can be expressed as:

Result: The integral evaluates to a combination of logarithmic functions and constants, specifically:

• log(sec x + tan x) + C
• Other terms involving cos x and cot x may also appear based on the integration process.

In summary, the integral can be solved through substitution, leading to a result that combines logarithmic expressions and constants. Each step in the process is crucial for arriving at the final answer.