Question

\( \int_{0}^{\frac{\sqrt{\pi}}{2}} 2x^3 \sin(x^2)\, dx = \)

• A. \( \frac{1}{\sqrt{2}}\left(1+\frac{\pi}{4}\right) \)
• B. \( \frac{1}{\sqrt{2}}\left(1-\frac{\pi}{4}\right) \)
• C. \( \frac{1}{\sqrt{2}}\left(\frac{\pi}{2}-1\right) \)
• D. \( \frac{1}{\sqrt{2}}\left(1-\frac{\pi}{2}\right) \)
• E. \( \frac{1}{\sqrt{2}}\left(\frac{\pi}{4}-1\right) \)

Hint:

Whenever you see \(x^n \sin(x^2)\), substitution \(t=x^2\) simplifies the integral immediately.

Answer 1

The Correct Option is B