Question

What is the value of $\int_{0}^{\pi} x |\cos x| \sin x \, dx$?

• A. $\frac{\pi}{2}$
• B. $\frac{\pi}{4}$
• C. $\pi$
• D. $\frac{\pi}{6}$

Hint:

Whenever the term $x$ is multiplied by a symmetric trigonometric expression over the interval $[0, \pi]$, King's property always simplifies the integral by converting $\int_0^\pi x f(x) \, dx$ to $\frac{\pi}{2} \int_0^\pi f(x) \, dx$.
This is a standard shortcut for competitive exams.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We use the property $\int_0^a f(x) dx = \int_0^a f(a-x) dx$.
Step 2: Applying the Property:
$I = \int_0^\pi x |\cos x| \sin x dx$. $I = \int_0^\pi (\pi-x) |\cos(\pi-x)| \sin(\pi-x) dx$ $I = \int_0^\pi (\pi-x) |-\cos x| \sin x dx = \int_0^\pi (\pi-x) |\cos x| \sin x dx$.
Step 3: Adding the Integrals:
$2I = \int_0^\pi (x + \pi - x) |\cos x| \sin x dx = \pi \int_0^\pi |\cos x| \sin x dx$. Using symmetry around $\pi/2$: $2I = 2\pi \int_0^{\pi/2} \cos x \sin x dx$ $I = \pi \int_0^{\pi/2} \sin x \cos x dx$.
Step 4: Final Evaluation:
$I = \pi \left[ \frac{\sin^2 x}{2} \right]_0^{\pi/2} = \pi [1/2 - 0] = \pi/2$.
Final Answer:
The value is π/2.