Question

\(\int\limits_0^{\frac{\pi }{2}} {co{s^2}\theta \,d\theta } \)To find the value of a given definite integral.

Short answer

\(\frac{\pi }{4}\)is the answer to a given question.

Step-by-step solution

Formula used

\({\cos ^2}\theta = \frac{{1 + \cos 2\theta }}{2}\)

Calculation of Definite Integral

\(\int\limits_0^{\frac{\pi }{2}} {{{\cos }^2}\theta = \int\limits_0^{\frac{\pi }{2}} {\frac{{1 + \cos 2\theta \,d\theta }}{2}} } \)

\( = \frac{1}{2}\int\limits_0^{\frac{\pi }{2}} {d\theta + \frac{1}{2}\int\limits_0^{\frac{\pi }{2}} {\cos 2\theta \,d\theta } } \)

\(\left( {\int {d\theta = \theta \,and\,\int {\cos 2\theta = \frac{{\sin 2\theta }}{2}} } } \right)\)

Divide \(\left( {\frac{\theta }{2} + \frac{{sin\,2\theta }}{4}} \right)_0^{\frac{\pi }{2}}\)Now putting the value of (upper limit) & (lower limit).

\( = \frac{\pi }{{2 \times 2}} + \frac{{\sin \,2 \times \frac{\pi }{2}}}{4} - 0\,\,\,\,(\sin \,\pi = 0)\)

\( = \frac{\pi }{4}\)\(\)

Hence, \(\frac{\pi }{4}\) is required Answer. \(\frac{\pi }{4}\) is find value of \(\int\limits_0^{\frac{\pi }{2}} {{{\cos }^2}\theta \,d\theta } \).