Evaluate the definite integral in Exercise: _ 0 ^ 2 ^ 2 x dx - Vidyadip

Question

Evaluate the definite integral in Exercise:
$\int\limits_{0}^{\frac{\pi}{2}}\cos^{2}\text{x}\ \text{dx}$

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Answer

$\text{Let}\ \text{I}=\int\limits_{0}^{\frac{\pi}{2}}\cos^{2}\text{x}\ \text{dx}$$\int\cos^{2}\text{x}\ \text{dx}=\int\bigg(\frac{1+\cos2\text{x}}{2}\bigg)\text{dx}=\frac{\text{x}}{2}+\frac{\sin2\text{x}}{4}=\frac{1}{2}\bigg(\text{x}+\frac{\sin2\text{x}}{2}\bigg)=\text{F}\text{(x)}$

By second fundamental theorem of calculus, we obtin

$\text{I}=\bigg[\text{F}\bigg(\frac{\pi}{2}\bigg)-\text{F}(0)\bigg]$

$=\frac{1}{2}\Bigg[\bigg(\frac{\pi}{2}-\frac{\sin\pi}{2}\bigg)-\bigg(0+\frac{\sin0}{2}\bigg)\Bigg]$

$=\frac{1}{2}\bigg[\frac{\pi}{2}+0-0-0\bigg]$

$=\frac{\pi}{4}$

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