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$\int_0^{1 / 2}|\sin 4 \pi x| d x=$
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1759 Upvotes
Verified Answer
The correct answer is:
$\frac{1}{\pi}$
$$
\begin{aligned}
& \text { Let } I=\int_0^{1 / 2}|\sin 4 \pi x| d x \\
& =2 \int_0^{1 / 4}|\sin 4 \pi x| d x=2 \int_0^{1 / 4} \sin 4 \pi x d x \\
& =2\left[\frac{-\cos 4 \pi x}{4 \pi}\right]_0^{1 / 4}=\frac{1}{2 \pi}[-\cos \pi+\cos 0] \\
& =\frac{1}{2 \pi}[1+1]=\frac{1}{\pi}
\end{aligned}
$$
\begin{aligned}
& \text { Let } I=\int_0^{1 / 2}|\sin 4 \pi x| d x \\
& =2 \int_0^{1 / 4}|\sin 4 \pi x| d x=2 \int_0^{1 / 4} \sin 4 \pi x d x \\
& =2\left[\frac{-\cos 4 \pi x}{4 \pi}\right]_0^{1 / 4}=\frac{1}{2 \pi}[-\cos \pi+\cos 0] \\
& =\frac{1}{2 \pi}[1+1]=\frac{1}{\pi}
\end{aligned}
$$
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