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

Question

Evaluate the definite integral in Exercise:$\int\limits_{0}^{\frac{\pi}{2}}\cos2\text{x}\ \text{dx}$

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Answer

$\text{Let}\ \text{I}=\int\limits_{0}^{\frac{\text{n}}{2}}\cos2\text{x}\ \text{dx}$$\int\cos2\text{x}\ \text{dx}=\bigg(\frac{\sin2\text{x}}{2}\bigg)=\text{F}\text{(x)}$
By second fundamental theorem of calculus, we obtain
$\text{I}=\text{F}\bigg(\frac{\pi}{2}\bigg)-\text{F}(0)$
$=\frac{1}{2}\bigg[\sin2\bigg(\frac{\pi}{2}\bigg)-\sin0\bigg]$
$=\frac{1}{2}[\sin\pi-\sin0]$
$=\frac{1}{2}[0-0]=0$

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