\frac{201 \int_{0}^{1} \left(\left(\right. \textrm{ } 1 - x^{4} \textrm{ } \left.\right)\right)^{\textrm{ } 50} \textrm{ } dx}{50 \int_{0}^{1} \left(\left(\right. \textrm{ } 1 - x^{4} \textrm{ } \left.\right)\right)^{\textrm{ } 49} \textrm{ } dx} equals

Engineering

Mathematics

Properties of Definite Integral

Question

$\frac{201 \int_{0}^{1} {(1 - x^{4})}^{50} dx}{50 \int_{0}^{1} {(1 - x^{4})}^{49} dx}$ equals

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$I_{n} = \int_{0}^{1} {(1 - x^{4})}^{n} dx = 4 n \int_{0}^{1} [1 - (1 - x^{4})] {(1 - x^{4})}^{n - 1} dx$

$I_{n} = 4 {nI}_{n - 1} - 4 {nI}_{n} \Rightarrow \frac{(4 n + 1) I_{n}}{{nI}_{n - 1}} = 4$