TS EAMCET · Maths · Definite Integration
By the definition of the definite integral, the value of \(\lim _{n \rightarrow \infty}\left(\frac{1^4}{1^5+n^5}+\frac{2^4}{2^5+n^5}+\frac{3^4}{3^5+n^5}+\ldots+\frac{n^4}{n^5+n^5}\right)\) is
- A (a) \(\log 2\)
- B \(\frac{1}{5} \log 2\)
- C \(\frac{1}{4} \log 2\)
- D \(\frac{1}{3} \log 2\)
Answer & Solution
Correct Answer
(B) \(\frac{1}{5} \log 2\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \begin{array}{l}\lim _{n \rightarrow \infty}\left(\frac{1^4}{1^5+n^5}+\frac{2^4}{2^5+n^5}+\frac{3^4}{3^5+n^5}+\ldots+\frac{n^4}{n^5+n^5}\right) \\ \qquad \sum_{r=0}^{r=n} \frac{r^4}{r^5+n^5}=\int_0^1 \frac{x^4}{1+x^5} d x \\ \text { Put } 1+x^5=t \Rightarrow 5…
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