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

Question

Evaluate the definite integral in Exercise:
$\int^{\pi}_{0}(\sin^{2}\frac{\text{x}}{2}-\cos^{2}\frac{\text{x}}{2})\text{dx}$

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Answer

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

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