Question

\(\int \tan \left( \tan^{-1} x \right) \, dx = ? \)

• A. \( \frac{x^2}{2} + k \)
• B. \( \frac{x}{2} + k \)
• C. \( x + k \)
• D. \( \log \sec (\tan^{-1} x) + k \)

Hint:

Whenever you encounter an inverse function inside a trigonometric function, simplify it first using the identity \(\tan(\tan^{-1} x) = x\).

Answer 1

The Correct Option is A

We are asked to evaluate the integral: \[ \int \tan \left( \tan^{-1} x \right) \, dx \] Step 1: Simplifying the integrand Since \(\tan^{-1} x\) is the inverse of the tangent function, we know that \(\tan (\tan^{-1} x) = x\). So, the integral becomes: \[ \int \tan \left( \tan^{-1} x \right) \, dx = \int x \, dx \] Step 2: Solving the integral Now, we integrate \(x\): \[ \int x \, dx = \frac{x^2}{2} + k \] Thus, the solution to the integral is: \[ \frac{x^2}{2} + k \] Therefore, the correct answer is: (A) \( \frac{x^2}{2} + k \).