Evaluate the following definite integrals: _ 0 ^ ( ^2 x 2 - ^2 x … - Vidyadip

Question

Evaluate the following definite integrals:
$\int_{0}^\limits{\pi}\Big(\sin^2\frac{\text{x}}{2}-\cos^2\frac{\text{x}}{2}\Big)\text{dx}$

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Answer

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

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