The value of $\int_0^{\pi/4} x \cos(x^2)dx$ correct to three decimal places (assuming that $\pi = 3.14$ ) is ______.

To evaluate the definite integral:

Use substitution: Let $u = x^2$, then $du = 2x\, dx \implies x\, dx = \frac{du}{2}$.

• Lower limit: when $x = 0$, $u = 0$.

• Upper limit: when $x = \frac{\pi}{4}$, $u = \left(\frac{\pi}{4}\right)^2 = \frac{\pi^2}{16}$.

Substituting these into the integral:

Given $\pi = 3.14$:

Evaluating the sine of $0.616225$ radians:

The value lies within the official key range of 0.27 to 0.30.