JEE Mains · Maths · STD 12 - 7.2 definite integral
The value of \(\mathrm{k} \in \mathbb{N}\) for which the integral \(I_n=\int_0^1\left(1-x^k\right)^n d x, n \in \mathbb{N} \text {, satisfies } 147 I_{20}=148 I_{21}\) is :
- A \(10\)
- B \(8\)
- C \(14\)
- D \(7\)
Answer & Solution
Correct Answer
(D) \(7\)
Step-by-step Solution
Detailed explanation
\( I_n=\int_0^1\left(1-x^k\right)^n \cdot 1 d x \) \( I_n=\left(1-x^k\right)^n \cdot x-n k \int_0^1\left(1-x^k\right)^{n-1} \cdot x^{k-1} \cdot d x \) \( I_n=n k \int_0^1\left[\left(1-x^k\right)^n-\left(1-x^k\right)^{n-1}\right] d x \) \( I_n=n k I_n-n k I_n \)…
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