JEE Mains · Maths · STD 12 - 7.2 definite integral
Let \(r_k=\frac{\int_0^1\left(1-x^7\right)^k d x}{\int_0^1\left(1-x^7\right)^{k+1} d x}, k \in N\). Then the value of \(\sum_{\mathrm{k}=1}^{10} \frac{1}{7\left(\mathrm{r}_{\mathrm{k}}-1\right)}\) is equal to ...........
- A \(69\)
- B \(47\)
- C \(65\)
- D \(37\)
Answer & Solution
Correct Answer
(C) \(65\)
Step-by-step Solution
Detailed explanation
\( I_K=\int 1 \cdot\left(1-x^7\right)^K d x \) \( I_K=\left.\left(1-x^7\right)^K x\right|_0 ^1+7 K \int_0^1\left(1-x^7\right)^{K-1} x^6 \cdot x d x \) \( I_K=-7 K \int_0^1\left(1-x^7\right)^{K-1}\left(\left(1-x^7\right)-1\right) d x \) \( I_K=-7 K \) \( I_K+7 K I_{K-1} \)…
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