Let $[t]$ denote the greatest integer less than or equal to $t$. Then the value of $\int_1^2|2 x-[3 x]| d x$ is

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Aicharya · Accepted Answer

$\begin{aligned} & \int_1^2|2 x-[3 x]| d x=\int_1^{\frac{4}{3}}|2 x-3| d x+\int_{\frac{4}{3}}^{\frac{5}{3}}|2 x-4| d x+\int_{\frac{5}{3}}^2|2 x-5| d x \ & =\int_1^{\frac{4}{3}}(3-2 x) d x+\int_{\frac{4}{3}}^{\frac{5}{3}}(4-2 x) d x+\int_{\frac{5}{3}}^2(5-2 x) d x \ & =\left[3 x-x^2ight]_1^{4 / 3}+\left[4 x-x^2ight]_{4 / 3}^{5 / 3}+\left[5 x-x^2ight]_{5 / 3}^2 \ & =3[x]_1^{4 / 3}+[4 x]_{4 / 3}^{5 / 3}+5\left[x^2ight]_{5 / 3}^2-\left[x^2ight]_1^2 \ & =(3+4+5) 	imes \frac{1}{3}-\left(2^2-1^2ight) \ & =4-3=1\end{aligned}$

Let $[t]$ denote the greatest integer less than or equal to $t$. Then the value of $\int_1^2|2 x-[3 x]| d x$…

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