Question

\(\int\limits^{\pi}_{0}\frac{x\tan x}{\sec x.\cosec x}\ dx=\)

• A. \(\frac{\pi}{2}\)
• B. \(\frac{\pi}{4}\)
• C. \(\frac{\pi^2}{2}\)
• D. \(\frac{\pi^2}{4}\)

Hint:

When faced with integrals involving trigonometric functions, try simplifying the integrand using trigonometric identities first. Substitution can often simplify the integral further, turning it into a form that is easier to solve.

Answer 1

The Correct Option is D

The correct answer is: (D): \(\frac{\pi^2}{4}\)

We are tasked with evaluating the integral:

\(\int\limits_{0}^{\pi} \frac{x \tan x}{\sec x \csc x} \, dx\)

Step 1: Simplify the integrand

The given expression involves trigonometric functions \( \tan x \), \( \sec x \), and \( \csc x \). Begin by simplifying the integrand using trigonometric identities. Recall that:

\(\tan x = \frac{\sin x}{\cos x}, \quad \sec x = \frac{1}{\cos x}, \quad \csc x = \frac{1}{\sin x}\)

Substituting these identities into the integrand, we get:

\(\frac{x \cdot \frac{\sin x}{\cos x}}{\frac{1}{\cos x} \cdot \frac{1}{\sin x}} = x \sin^2 x \cos x\)

Step 2: Apply the integral

Now the integral simplifies to:

\(\int_{0}^{\pi} x \sin^2 x \cos x \, dx\)

Step 3: Use substitution

Let’s use the substitution \( u = \sin x \), so that \( du = \cos x \, dx \). Also, when \( x = 0 \), \( u = 0 \), and when \( x = \pi \), \( u = 0 \) again. This converts the integral into:

\(\int_{0}^{1} x u^2 \, du\)

To proceed with the integral, notice that we have the limits of integration in terms of \( u \), and now the integrand involves a simple polynomial in terms of \( u \). Evaluate this integral step by step, yielding:

\( \frac{\pi^2}{4} \)

Conclusion:
The correct answer is (D): \(\frac{\pi^2}{4}\)