GATE CS 2018 Q.26 (CS Q.16): Calculus

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

Your answer

Official answer and solution

Solution

Use substitution $u = x^2$ to evaluate the definite integral.

Consider the integral:

I = \int_0^{\pi/4} x \cos(x^2) dx

Let $u = x^2$ , so $du = 2x\,dx$ , giving $x\,dx = \frac{du}{2}$ .

Lower limit: when $x = 0$ , $u = 0$ .
Upper limit: when $x = \frac{\pi}{4}$ , $u = \left(\frac{\pi}{4}\right)^2 = \frac{\pi^2}{16}$ .

Substitute into the integral:

I = \int_0^{\pi^2/16} \cos(u) \frac{du}{2} = \left[ \frac{\sin(u)}{2} \right]_0^{\pi^2/16} = \frac{\sin\left(\frac{\pi^2}{16}\right)}{2}

Using $\pi = 3.14$ :

\frac{\pi^2}{16} = \frac{3.14^2}{16} = \frac{9.8596}{16} = 0.616225\text{ radians}

\sin(0.616225\text{ rad}) \approx 0.57796

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

Rounding to three decimal places yields 0.289.

Final answer

0.289

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