Inverse Trigonometric Functions
Derivatives
Premium Question
Grade 12
Question:
$f'(x) = \tan^{-1}(\sec x + \tan x)$ for $x \in (-\pi/2, \pi/2)$ and $f(0) = 0$. Then $f(1)$ equals:
(1) $\frac{\pi + 1}{4}$
(2) $\frac{\pi + 2}{4}$
(3) $\frac{1}{4}$
(4) $\frac{\pi - 1}{4}$
Step-by-Step Solution
Key Concept: Write $\sec x + \tan x = \frac{1 + \sin x}{\cos x}$. Use the half-angle identity to show $f'(x) = \frac{\pi}{4} + \frac{x}{2}$, then integrate.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)