Evaluate the following integrals: ^ 4 _ - 4 | |dx

^ 4 _ - 4 | |dxWe know that, | |= - ,&- 4 0 ,&0<x 4 I= ^ 4 _ - 4 | |dx = ^ 0_ - 4 - dx+ ^ 4 _0 dx = [ ]^0_ - 4 - [ ]^ - 4 _0 =1-1 2 -1 2 +1 =2-2 2 =2-2

Evaluate the following integrals: ^ 4 _ - 4

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Question

Evaluate the following integrals:$\int^\limits{\frac{\pi}{4}}_{\frac{-\pi}{4}}|\sin\text{x}|\text{dx}$

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Answer

$\int^\limits{\frac{\pi}{4}}_{\frac{-\pi}{4}}|\sin\text{x}|\text{dx}$We know that,
$|\sin\text{x}|=\begin{cases}-\sin\text{x},&-\frac{\pi}{4}\leq\text{x}\leq0\\\sin\text{x},&0<\text{x}\leq\frac{\pi}{4}\end{cases}$
$\therefore\ \text{I}=\int^\limits{\frac{\pi}{4}}_{\frac{-\pi}{4}}|\sin\text{x}|\text{dx}$
$\Rightarrow\text{I}=\int^\limits0_{-\frac{\pi}{4}}-\sin\text{x dx}+\int\limits^{\frac{\pi}{4}}_0\sin\text{x dx}$
$\Rightarrow\text{I}=\big[\cos\text{x}\big]^0_{\frac{-\pi}{4}}-\big[\cos\text{x}\big]^{\frac{-\pi}{4}}_0$
$\Rightarrow\text{I}=1-\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}+1$
$\Rightarrow\text{I}=2-\frac{2}{\sqrt{2}}$
$\Rightarrow\text{I}=2-\sqrt{2}$

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