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 Question:
We must evaluate a highly complex definite integral with reciprocal limits ($1/2$ and $2$). The presence of $x - 1/x$ strongly hints at a reciprocal substitution trick.

Step 2: Detailed Explanation:
Let the given integral be:
$I = \int_{1/2}^{2} \frac{1}{x} \csc^{101} \left( x - \frac{1}{x} \right) dx$ --- (Equation 1)
Apply the substitution $x = \frac{1}{t}$.
Differentiating both sides: $dx = -\frac{1}{t^2} dt$.
Now, update the limits of integration:
When $x = 1/2 \implies t = 2$.
When $x = 2 \implies t = 1/2$.
Substitute these into the integral:
$I = \int_{2}^{1/2} \frac{1}{(1/t)} \csc^{101} \left( \frac{1}{t} - t \right) \left(-\frac{1}{t^2}\right) dt$
$I = \int_{2}^{1/2} t \cdot \csc^{101} \left( \frac{1}{t} - t \right) \left(-\frac{1}{t^2}\right) dt$
$I = \int_{2}^{1/2} -\frac{1}{t} \csc^{101} \left( \frac{1}{t} - t \right) dt$
Use the property of definite integrals that swapping the upper and lower limits changes the sign of the integral ($\int_a^b = -\int_b^a$):
$I = \int_{1/2}^{2} \frac{1}{t} \csc^{101} \left( \frac{1}{t} - t \right) dt$
Now, apply the trigonometric identity that Cosecant is an odd function ($\csc(-\theta) = -\csc(\theta)$). Therefore, $\csc(1/t - t) = \csc(-(t - 1/t)) = -\csc(t - 1/t)$.
Since it is raised to an odd power (101), the negative sign completely pulls out:
$I = \int_{1/2}^{2} \frac{1}{t} \left[ -\csc \left( t - \frac{1}{t} \right) \right]^{101} dt$
$I = -\int_{1/2}^{2} \frac{1}{t} \csc^{101} \left( t - \frac{1}{t} \right) dt$
Since the variable of integration is a dummy variable, we can replace $t$ with $x$:
$I = -\int_{1/2}^{2} \frac{1}{x} \csc^{101} \left( x - \frac{1}{x} \right) dx$
Notice that the right side is exactly the negative of our original integral $I$ (from Equation 1):
$I = -I$
Solve for $I$:
$2I = 0 \implies I = 0$.

Step 3: Final Answer:
The integral evaluates to 0, matching option (a).