Question

\(\int e^{\tan x}(\tan^{7}x+5\tan^{6}x+\tan^{5}x+5\tan^{4}x)\,dx=\)

• A. \(e^{\tan x}\frac{\tan^{8}x}{8}+c\)
• B. \(e^{\tan x}(\tan^{5}x)+c\)
• C. \(e^{\tan x}\frac{\tan^{6}x}{6}+c\)
• D. \(e^{\tan x}(\tan^{7}x)+c\) \bigskip

Hint:

Look for \(e^t \cdot P(t)\) → check product rule reverse.

Answer 1

The Correct Option is A

Concept: Let \(t=\tan x\), then \(dt=\sec^2 x dx\).

Step 1: Structure recognition.
Expression becomes: \[ e^t(t^7+5t^6+t^5+5t^4) \]

Step 2: Factor pattern.
\[ = e^t \cdot t^4(t^3+5t^2+t+5) \] This matches derivative of: \[ e^t \cdot \frac{t^8}{8} \] \[ \boxed{(A)} \]