Question

$\int_{1/2}^{2} \frac{1}{x} \csc^{101} \left( x - \frac{1}{x} \right) dx = $ ______.

• A. 0
• B. 1
• C. $\frac{1}{4}$
• D. $\frac{101}{2}$

Hint:

Whenever you see reciprocal integration limits (like $a$ and $1/a$) combined with an odd function argument like $(x - 1/x)$, substituting $x = 1/t$ will almost always flip the sign and prove that the integral is exactly 0!

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Use the substitution property of definite integrals, specifically $x = 1/t$, for integrals with reciprocal limits.

Step 2: Formula Application:
Let $x = \frac{1}{t}$, then $dx = -\frac{1}{t^2} dt$. Limits change from $[1/2, 2]$ to $[2, 1/2]$.

Step 3: Explanation:
$I = \int_{2}^{1/2} t \cdot \csc^{101} (1/t - t) \cdot (-\frac{1}{t^2}) dt$ $I = \int_{1/2}^{2} \frac{1}{t} \csc^{101} [-(t - 1/t)] dt$ Since $\csc(- \theta) = -\csc \theta$ and the power (101) is odd: $I = \int_{1/2}^{2} \frac{1}{t} [-\csc^{101} (t - 1/t)] dt = -I$. $2I = 0 \implies I = 0$.

Step 4: Final Answer:
The value of the integral is 0.