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: Understanding the Concept
This integral can be solved by first simplifying the integrand using trigonometric identities and then recognizing that the resulting expression is the derivative of a product of functions.
Step 2: Key Formula or Approach
1. Use the Pythagorean identity $1+\tan^2 x = \sec^2 x$ to simplify the integrand. 2. Expand the simplified expression. 3. Recognize the expanded form as the result of applying the product rule of differentiation, $\frac{d}{dx}(uv) = u'v+uv'$. 4. Integrate the recognized derivative.
Step 3: Detailed Explanation
The integral is $I = \int(1+\tan^2 x)(1+2x\tan x)dx$. 1. Simplify the integrand: Using the identity $1+\tan^2 x = \sec^2 x$, the integrand becomes: \[ \sec^2 x(1+2x\tan x) \] Expand this expression: \[ \sec^2 x + 2x\tan x\sec^2 x \] 2. Recognize the derivative form: The expression looks like the result of a product rule. Let's consider the function $f(x) = x \sec^2 x$. Let's find its derivative using the product rule, where $u=x$ and $v=\sec^2 x$. \[ f'(x) = \frac{d}{dx}(x) \cdot \sec^2 x + x \cdot \frac{d}{dx}(\sec^2 x) \] We need the derivative of $\sec^2 x$. Using the chain rule: \[ \frac{d}{dx}(\sec^2 x) = 2\sec x \cdot \frac{d}{dx}(\sec x) = 2\sec x \cdot (\sec x \tan x) = 2\sec^2 x \tan x \] Substituting this back into the product rule: \[ f'(x) = 1 \cdot \sec^2 x + x \cdot (2\sec^2 x \tan x) = \sec^2 x + 2x\sec^2 x \tan x \] This is exactly the integrand we obtained after simplification. 3. Evaluate the integral: Since the integrand is the derivative of $x\sec^2 x$, the integral is simply the function itself. \[ I = \int \frac{d}{dx}(x\sec^2 x) dx = x\sec^2 x + c \] Step 4: Final Answer
The integral evaluates to $x\sec^2 x + c$.