Question

Solve this question..?​

Answer 1

Step-by-step explanation:

Let $\rm\:I=\int^{\frac{\pi}{2}}_{0}\cos^{2}(x)\:dx\:\:....(i)$

$\implies\rm\:I=\int^{\frac{\pi}{2}}_{0}\cos^{2} \bigg( \frac{\pi}{2} + 0 - x \bigg)\:dx$

$\implies\rm\:I=\int^{\frac{\pi}{2}}_{0}\cos^{2} \bigg( \frac{\pi}{2} - x \bigg)\:dx$

$\implies\rm\:I=\int^{\frac{\pi}{2}}_{0}\sin^{2}( x)\:dx \: \: ....(ii)$

Adding (i) and (ii),

$\implies\rm\:2I=\int^{\frac{\pi}{2}}_{0}\cos^{2}( x)\:dx + \int^{\frac{\pi}{2}}_{0}\sin^{2}( x)\:dx$

$\implies\rm\:2I=\int^{\frac{\pi}{2}}_{0}(\cos^{2}( x) + \sin^{2}( x))\:dx$

$\implies\rm\:2I=\int^{\frac{\pi}{2}}_{0}(1)\:dx$

$\implies\rm\:2I=\int^{\frac{\pi}{2}}_{0}\:dx$

$\implies\rm\:2I= \bigg[ x \bigg]^{ \frac{\pi}{2} }_{0}$

$\implies\rm\:2I= \bigg[ \frac{\pi}{2} - 0 \bigg]$

$\implies\rm\:2I= \frac{\pi}{2}$

$\implies\rm\:I= \frac{\pi}{4}$