Question

Evaluate: $$ \int_0^1 x \sin^{-1} x \, dx $$

• A. \( \frac{\pi}{8} \)
• B. \( \frac{\pi}{4} \)
• C. \( \frac{\pi}{12} \)
• D. \( \frac{\pi}{3} \)

Hint:

Use integration by parts and trigonometric substitution.

Answer 1

The Correct Option is A

Use integration by parts: Let \[ u = \sin^{-1} x, \quad dv = x \, dx \] \[ du = \frac{1}{\sqrt{1 - x^2}} dx, \quad v = \frac{x^2}{2} \] Then, \[ \int_0^1 x \sin^{-1} x \, dx = \left. \frac{x^2}{2} \sin^{-1} x \right|_0^1 - \int_0^1 \frac{x^2}{2 \sqrt{1 - x^2}} dx \] Evaluate the remaining integral using substitution \( x = \sin \theta \). Final result: \[ \frac{\pi}{8} \]