माना I_{1}=int_{0}^{n}[x] d x और I_{2}=int_{0 | vedclass

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Question

(D) हमारे पास $I_{1} = \int_{0}^{n} [x] dx = \sum_{k=0}^{n-1} \int_{k}^{k+1} k dx = \sum_{k=0}^{n-1} k(k+1-k) = \sum_{k=0}^{n-1} k = \frac{(n-1)n}{2}$

Vedclass · Accepted Answer

$n-1$

Vedclass · Answer

(D) हमारे पास $I_{1} = \int_{0}^{n} [x] dx = \sum_{k=0}^{n-1} \int_{k}^{k+1} k dx = \sum_{k=0}^{n-1} k(k+1-k) = \sum_{k=0}^{n-1} k = \frac{(n-1)n}{2}$ है।अब, $I_{2} = \int_{0}^{n} \{x\} dx$. चूँकि $\{x\} = x - [x]$, इसलिए $I_{2} = \int_{0}^{n} x dx - \int_{0}^{n} [x] dx$ है।$I_{2} = \left[ \frac{x^2}{2} \right]_{0}^{n} - I_{1} = \frac{n^2}{2} - \frac{n(n-1)}{2} = \frac{n^2 - n^2 + n}{2} = \frac{n}{2}$ है।अतः, $\frac{I_{1}}{I_{2}} = \frac{\frac{n(n-1)}{2}}{\frac{n}{2}} = n-1$।

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