Definite Integration
General
Grade 12
Question:
Let $I_n = \int_0^{\pi/4} \tan^n x dx$ ($n > 1$ and is an integer). Show that $\frac{1}{2(n+1)} < I_n < \frac{1}{2(n-1)}$.
Step-by-Step Solution
Key Concept: General
$I_n = \int_0^{\pi/4} \tan^n x dx = \int_0^{\pi/4} \tan^2 x \cdot \tan^{n-2} x dx = \int_0^{\pi/4} (\sec^2 x - 1) \tan^{n-2} x dx = \int_0^{\pi/4} (\sec^2 x \cdot \tan^{n-2} x) dx - \int_0^{\pi/4} \tan^{n-2} x dx \Rightarrow I_n + I_{n-2} = \int_0^{\pi/4} \tan^{n-2} x d(\tan x) = \left. \frac{\tan^{n-1} x}{n-1} \right|_0^{\pi/4} = \frac{1}{n-1}$
Correct Answer: A