Let β(m, n)=int 0 1 x m-1 (1-x) n-1 d x, m, n 0 . If int 0 1 (1-x 10 ) 20 d x=a × β(b, c) , then 100(a+b+x) equals

Mathematics · Definite Integration

JEE Main 2024 — 5 April, Shift 2 — Question 17

Let $\beta(m, n)=\int_{0}^{1} x^{m-1}(1-x)^{n-1} d x, m, n>0$ . If $\int_{0}^{1}\left(1-x^{10}\right)^{20} d x=a \times \beta(b, c)$ ,

then $100(a+b+x)$ equals $\qquad$

$\quad I=\int_{0}^{1} 1 .\left(1-x^{10}\right)^{20} d x$ $\mathrm{x}^{10}=\mathrm{t}$

$\mathrm{x}=\mathrm{t}^{1 / 10}$

$\mathrm{dx}=\frac{1}{10}(\mathrm{t})^{-9 / 10} \mathrm{dt}$

$I=\int_{0}^{1}(1-t)^{20} \frac{1}{10}(t)^{-9 / 10} d t$

$I=\frac{1}{10} \int_{0}^{1} \mathrm{t}^{-9 / 10}(1-\mathrm{t})^{20} d t$

$\mathrm{a}=\frac{1}{10} \quad \mathrm{~b}=\frac{1}{10} \quad \mathrm{c}=21$

Answer key and solution verified before publishing.

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