Definite Integration
King's Rule — Sum of Square Roots
nta_pyq_2026_jan
Grade 12
Question:
The value of $\displaystyle\sum_{r=1}^{20}\left(\left|\sqrt{\pi\left(\int_0^r x|\sin\pi x|\,dx\right)}\right|\right)$ is _____
Step-by-Step Solution
Key Concept: Let $I_r=\int_0^r x|\sin\pi x|\,dx$. Apply King's rule to get $2I_r=r\int_0^r|\sin\pi x|\,dx$. For integer $r$: $\int_0^r|\sin\pi x|\,dx=\frac{2r}{\pi}$. So $I_r=\frac{r^2}{\pi}$.
$I_r=r^2/\pi$. Sum $=\sum_{r=1}^{20}r=210$.
Correct Answer: 210