Question

Evaluate: \(\rm \int_{0}^{\pi/2}\cos^2x\ dx\)
1. \(\frac\pi2\)
2. \(\frac\pi3\)
3. \(\frac\pi4\)
4. \(\frac\pi6\)

Answer 1

Correct Answer - Option 3 : \(\frac\pi4\)

Concept:

Definite Integral:

If ∫ f(x) dx = g(x) + C, then \(\rm \int_a^b f(x)\ dx = [ g(x)]_a^b\) = g(b) - g(a).

Trigonometric Identities:

cos 2x = 2 cos² x - 1

Calculation:

Let I = \(\rm \int_{0}^{\pi/2}\cos^2x\ dx\)

⇒ I = \(\rm \int_{0}^{\pi/2}\frac{\cos 2x+1}{2}\ dx\)

⇒ I = \(\rm \frac{1}{2}\left[\int_{0}^{\pi/2}\cos 2x\ dx+\int_{0}^{\pi/2}1\ dx\right]\)

⇒ I = \(\rm \frac{1}{2}\left[\frac{\sin 2x}{2}+x\right]_0^{\pi/2}\)

⇒ I = \(\rm \frac{1}{2}\left[\left(\frac{\sin \pi}{2}+\frac{\pi}{2}\right)-\left(\frac{\sin 0}{2}+0\right)\right]\)

⇒ I = \(\frac\pi4\).