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  4. undersett arrow 0 textLim ∫ limits02 π (| sin (x+t)- sin x|/|t|) d x equals

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Q. $\underset{t \rightarrow 0}{\text{Lim}} \int\limits_0^{2 \pi} \frac{|\sin (x+t)-\sin x|}{|t|} d x$ equals

Integrals

Solution:

$I=\underset{t \rightarrow 0}{\text{Lim}} \int\limits_0^{2 \pi}\left|\frac{\sin (x+t)-\sin x}{t}\right| d x=\int\limits_0^{2 \pi}\left(\underset{t \rightarrow 0}{\text{Lim}}\left|\frac{2 \cos \left(x+\frac{t}{2}\right) \sin \frac{t}{2}}{t}\right|\right) d x$
$=\int\limits_0^{2 \pi} \underset{t \rightarrow 0}{\operatorname{Lim}}\left|\cos \left(x+\frac{t}{2}\right) \frac{\sin \frac{t}{2}}{\frac{t}{2}}\right|=\int\limits_0^{2 \pi}|\cos x| d x=4$