JEE Mains · Maths · STD 12 - 7.2 definite integral
If \(f: R \rightarrow R\) is given by \(f(x)=x+1\), then the value of \(\lim _{n \rightarrow \infty} \frac{1}{n}\left[f(0)+f\left(\frac{5}{n}\right)+f\left(\frac{10}{n}\right)+\ldots+f\left(\frac{5(n-1)}{n}\right)\right]\), is:
- A \(\frac{3}{2}\)
- B \(\frac{7}{2}\)
- C \(\frac{5}{2}\)
- D \(\frac{1}{2}\)
Answer & Solution
Correct Answer
(B) \(\frac{7}{2}\)
Step-by-step Solution
Detailed explanation
\(I=\sum_{r=0}^{n-1} f\left(\frac{5 r}{n}\right) \frac{1}{n}\) \(I=\int_{0}^{1} f(5 x) \,d x\) \(I=\int_{0}^{1}(5 x+1) \,d x\) \(I=\int_{0}^{1}\left[\frac{5 x^{2}}{2}+x\right]_{0}^{1}\) \(I=\frac{5}{2}+1=\frac{7}{2}\)
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