Question

Evaluate the integral: \[ \int \sec^5 x \tan x \, dx \] The correct answer is:

• A. \( 5 \tan^5 x + c \)
• B. \( \frac{1}{5} \sec^5 x + c \)
• C. \( 5 \log \left| \cos x \right| + c \)
• D. \( \tan^5 x + c \)

Hint:

To solve integrals of the form \(\int \sec^n x \tan x \, dx\), recognize that the integral is the derivative of \(\sec^{n+1} x\), and adjust accordingly.

Answer 1

The Correct Option is A

The integral can be simplified by recognizing that \(\sec^5 x \tan x\) can be written as the derivative of \(\sec^5 x\): \[ \frac{d}{dx} \left( \sec^5 x \right) = 5 \sec^4 x \cdot \sec x \tan x. \] Thus, the integral becomes: \[ \int \sec^5 x \tan x \, dx = \frac{1}{5} \sec^5 x + c. \] Therefore, the correct answer is \( (B) \).