GATE CS 2018 Q16 — Maxima and Minima, Mean Value Theorem, Integration

The value of

\int^{\pi/4} _0 x \cos(x^2) dx

correct to three decimal places (assuming that

\pi = 3.14

) is ____

View worked solution

Answer: 0.288 : 0.289

To evaluate the definite integral:

I = \int^{\pi/4} _0 x \cos(x^2) dx

We can use the method of substitution. Let $u = x^2$ . Then, the differential is:

du = 2x \, dx \implies x \, dx = \frac{du}{2}

Next, we update the limits of integration:

When $x = 0$ , $u = 0^2 = 0$ .
When $x = \frac{\pi}{4}$ , $u = \left(\frac{\pi}{4}\right)^2 = \frac{\pi^2}{16}$ .

Substituting these into the integral:

I = \int_{0}^{\pi^2/16} \cos(u) \frac{du}{2}

I = \frac{1}{2} \Big[ \sin(u) \Big]_{0}^{\pi^2/16}

I = \frac{1}{2} \sin\left(\frac{\pi^2}{16}\right)

Given that $\pi = 3.14$ :

\pi^2 = (3.14)^2 = 9.8596

\frac{\pi^2}{16} = \frac{9.8596}{16} \approx 0.616225 \text{ radians}

Now, we compute the sine of $0.616225$ radians:

\sin(0.616225) \approx 0.57796

Finally, we calculate the value of the integral $I$ :

I = \frac{1}{2} \times 0.57796 \approx 0.28898

Rounding to three decimal places, the value is approximately 0.289 (or 0.288 if intermediate values are rounded differently).

Free practice · Your answers on this page are not saved.

Published 10 Oct 2026

Related questions