The value of ( r = 1 20 | ( 0 r x sin ( x ) d x ) | ) is [2026]

Question

The value of $\sum_{r = 1}^{20} (| \sqrt[]{π (\int_{0}^{r} x | \sin π x | d x)} |)$ is _______. [2026]

Smart Achievers · Accepted Answer

(210)

$Let I_{r} = \int_{0}^{r} x | \sin π x | d x . . . (1)$

$Apply King Property$

$= \int_{0}^{r} (r - x) | \sin π x | d x . . . (2)$

$By (1) + (2)$

$2 I_{r} = \int_{0}^{r} r | \sin π x | d x \Rightarrow I_{r} = \frac{r}{2} \int_{0}^{r} | \sin π x | d x$

$I_{1} = \frac{1}{2} \int_{0}^{1} | \sin π x | d x = \frac{1}{2 π} \int_{0}^{π} | \sin t | d t = \frac{1}{2 π} (2)$

$I_{2} = \frac{2}{2} \int_{0}^{2} | \sin π x | d x = \frac{2}{2 π} \int_{0}^{2 π} | \sin t | d t = \frac{2}{2 π} (4)$

$S = \sqrt{π \cdot \frac{1}{2 π} \cdot 2} + \sqrt{π \cdot \frac{2}{2 π} \cdot 4} + \sqrt{π \cdot \frac{3}{2 π} \cdot 6} + \dots + \sqrt{π \cdot \frac{20}{2 π} \cdot (2 \cdot 20)}$

$= 1 + 2 + 3 + \dots + 20$

$= \frac{20 \times 21}{2} = 210$

Solution

Q.
The value of $\sum_{r = 1}^{20} (| \sqrt[]{π (\int_{0}^{r} x | \sin π x | d x)} |)$ is _______. [2026]

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Solution

Q. The value of ∑r=120(|π(∫0rx|sinπx|dx)|) is _______. [2026]

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Q.
The value of $\sum_{r = 1}^{20} (| \sqrt[]{π (\int_{0}^{r} x | \sin π x | d x)} |)$ is _______. [2026]