Question

$\int e^{\tan x}(\sec ^2 x+\sec ^3 x \sin x) d x=$

• A. $\tan x \cdot e^{\tan x}+c$
• B. $(1+\tan x) e^{\tan x}+c$
• C. $\sec x \cdot e^{\tan x}+c$
• D. $e^{\tan x}+\tan x+c$

Hint:

If you see $e^{f(x)}$ and a term involving $f'(x)$, substitution is usually the fastest path to the solution.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need to evaluate the integral of an exponential-trigonometric function.

Step 2: Key Formula or Approach:
Recognize the form $\int e^f(x) (f'(x) + g(x)) dx$ or use substitution.

Step 3: Detailed Explanation:
$\int e^{\tan x} (\sec^2 x + \sec^3 x \sin x) dx = \int e^{\tan x} \sec^2 x (1 + \sec x \sin x) dx$.
Since $\sec x \sin x = \frac{1}{\cos x} \sin x = \tan x$, the expression becomes:
$\int e^{\tan x} \sec^2 x (1 + \tan x) dx$.
Let $u = \tan x$, then $du = \sec^2 x dx$.
The integral becomes $\int e^u (1 + u) du = \int (e^u + u e^u) du$.
By parts on $\int u e^u du$: $u e^u - \int e^u du = u e^u - e^u$.
So, $\int (e^u + u e^u) du = e^u + u e^u - e^u = u e^u + c$.
Substituting back $u = \tan x$: $\tan x \cdot e^{\tan x} + c$.

Step 4: Final Answer:
The integral is $\tan x \cdot e^{\tan x} + c$, which is option (A).