Question

$\int(1+\tan^2 x)(1+2x\tan x)dx =$

• A. $x\sec x + c$
• B. $x\tan^2 x + c$
• C. $x\sec^2 x + c$
• D. $x\tan x + c$

Hint:

When an integrand is a sum of terms, especially after simplification, always check if it matches the form of a product rule derivative, $u'v+uv'$. This pattern recognition can turn a potentially difficult integral into a trivial one.

Answer 1

The Correct Option is C

Step 1: Simplify the integrand We know \( 1 + \tan^2 x = \sec^2 x \). So the integral is \( I = \int \sec^2 x (1 + 2x \tan x) dx \). \( I = \int (\sec^2 x + 2x \sec^2 x \tan x) dx \).
Step 2: Identify the derivative form Check the derivative of \( x \sec^2 x \): \[ \frac{d}{dx} (x \sec^2 x) = 1 \cdot \sec^2 x + x \cdot \frac{d}{dx}(\sec^2 x) \] \[ = \sec^2 x + x \cdot (2 \sec x \cdot \sec x \tan x) \] \[ = \sec^2 x + 2x \sec^2 x \tan x \] \[ = \sec^2 x (1 + 2x \tan x) \]
Step 3: Conclusion Since the integrand is exactly the derivative of \( x \sec^2 x \), the integral is: \[ I = x \sec^2 x + c \]