Evaluate the definite integral in Exercise: _0^ 4 2x dx - Vidyadip

Question

Evaluate the definite integral in Exercise:$\int\limits_0^{\frac{\pi}{4}}\sin2\text{x}\ \text{dx}$

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Answer

$\text{Let}\ \text{I}=\int\limits_{0}^{\frac{\text{x}}{4}}\sin2\text{x} \ \text{dx}$$\int\sin2\text{x}\ \text{dx}=\bigg(\frac{-\cos2\text{x}}{2}\bigg)=\text{F}\text{(x)}$
By second fundamental theorem of calculus, we obtain
$\text{I}=\text{F}\bigg(\frac{\pi}{4}\bigg)-\text{F}(0)$
$=-\frac{1}{2}\pi\bigg[\cos2\bigg(\frac{\pi}{4}\bigg)-\cos0\bigg] $
$=-\frac{1}{2}\bigg[\cos\bigg(\frac{\pi}{2}\bigg)-\cos0\bigg]$
$=-\frac{1}{2}[0-1]$
$=\frac{1}{2}$

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