Evaluate the following integrals: ^ 2 _ - 2 |x|+ |x| dx - Vidyadip

Question

Evaluate the following integrals:
$\int^\limits{\frac{\pi}{2}}_{\frac{-\pi}{2}}\big\{\sin|\text{x}|+\cos|\text{x}|\big\}\text{dx}$

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Answer

We know,
$\text{I}=\int^\limits{\frac{\pi}{2}}_{\frac{-\pi}{2}}\big\{\sin|\text{x}|+\cos|\text{x}|\big\}\text{dx}$
Let $\text{f(x)}=\sin|\text{x}|+\cos|\text{x}|$
Then, $\text{f(x)}=\text{f(-x)}$
$\therefore\ \text{f(x)}$ is an even function.
So,
$\text{I}=\int^\limits{\frac{\pi}{2}}_{\frac{-\pi}{2}}\big\{\sin|\text{x}|+\cos|\text{x}|\big\}\text{dx}$
$=2\int\limits^{\frac{\pi}{2}}_0(\sin\text{x}+\cos\text{x})\text{dx}$
$\\=2\big[-\cos\text{x}+\sin\text{x}\big]^{\frac{\pi}{2}}_0$
$=4$

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