Evaluate-1-0-1-1-1-x-2-100-201-xdx-0-1-1-1-x-2-100-202-xdx-2-0-1-1-x-200-201-dx-0-1-1-x-200-202-dx-

Question Number 56280 by rahul 19 last updated on 13/Mar/19

E v a l u a t e :

1) \frac{\int_{0}^{1_{}} {(1 - {(1 - x^{2})}^{100})}^{201} . x d x}{\int_{0}^{1} {(1 - {(1 - x^{2})}^{100})}^{202} . x d x} = ?

2) \frac{\int_{0}^{1} {(1 - x^{200})}^{201} d x}{\int_{0}^{1} {(1 - x^{200})}^{202} d x} = ?

Commented by rahul 19 last updated on 13/Mar/19

A n s :

1) \frac{20201}{20200} 2) \frac{40401}{40400} .

Commented by tanmay.chaudhury50@gmail.com last updated on 13/Mar/19

r a h u l p l s c h e c k t h e a n s w e r

Answered by tanmay.chaudhury50@gmail.com last updated on 13/Mar/19

2) t = x^{200} d t = 200 x^{199} d x

d t = 200 \times {(t^{\frac{1}{200}})}^{199} d x

d x = \frac{d t}{200 \times t^{\frac{199}{200}}}

\frac{\int_{0}^{1} {(1 - x^{200})}^{201} d x}{\int_{0}^{1} {(1 - x^{200})}^{202} d x}

\frac{\int_{0}^{1} {(1 - t)}^{201} \times \frac{d t}{200 \times t^{\frac{199}{200}}}}{\int_{0}^{1} {(1 - t)}^{202} \times \frac{d t}{200 \times t^{\frac{199}{200}}}}

\frac{\int_{0}^{1} {(1 - t)}^{201} \times t^{\frac{1}{200} - 1} d t}{\int_{0}^{1} {(1 - t)}^{202} \times t^{\frac{1}{200} - 1} d t}

n o w l o o k f o r m u l a β f u n c t i o n

β (m, n) = \int_{0}^{1} x^{m - 1} {(1 - x)}^{n - 1} d x

= \frac{⌈ (m) ⌈ (n)}{⌈ (m + n)}

= \frac{\frac{⌈ (202) ⌈ (\frac{1}{200})}{⌈ (202 + \frac{1}{200})}}{\frac{⌈ (203) ⌈ (\frac{1}{200})}{⌈ (203 + \frac{1}{200})}}

n o w ⌈ (n + 1) = n!

= \frac{\frac{201!}{(201 + \frac{1}{200})!}}{\frac{202!}{(202 + \frac{1}{200})!}}

= \frac{201! \times (202 + \frac{1}{200})!}{202! \times (201 + \frac{1}{200})!}

= \frac{1}{202} \times (202 + \frac{1}{200}) \times (201 + \frac{1}{200})! \times \frac{1}{(201 + \frac{1}{200})!}

= 1 + \frac{1}{200 \times 202}

= l

Commented by rahul 19 last updated on 13/Mar/19

I t w i l l b e

= \frac{1}{202} \times (202 + \frac{1}{200}) = 1 + \frac{1}{202 \times 200}

= \frac{40401}{40400} A n s .!

T h a n k y o u S i r!

Commented by tanmay.chaudhury50@gmail.com last updated on 13/Mar/19

t h a n k y o u r a h u l \dots s o m e t i m e s i l o s s c o n s i s t s n c y \dots

Answered by tanmay.chaudhury50@gmail.com last updated on 13/Mar/19

1) \frac{\int_{0}^{1} {1 - {(1 - x^{2})}^{a}}^{2 a + 1} . x d x}{\int_{0}^{1} {1 - {(1 - x^{2})}^{a}}^{2 a + 2} . x d x} a = 100

t = 1 - x^{2} d t = - 2 x d x

\frac{\frac{1}{2} \int_{1}^{0} {(1 - t^{a})}^{2 a + 1} \times - d t}{\frac{1}{2} \int_{1}^{0} {(1 - t^{a})}^{2 a + 2} \times - d t}

\frac{\int_{0}^{1} {(1 - t^{a})}^{2 a + 1} d t}{\int_{0}^{1} {(1 - t^{a})}^{2 a + 2} d t}

\frac{\int_{0}^{1} {(1 - t^{100})}^{201} d t}{\int_{0}^{1} {(1 - t^{100})}^{202} d t}

l o o k s a m e t y p e q u e s t i o n o f q u e s t i o n n o 2

b u t s o l v i n g o t h e r w a y

l e t t^{100} = {s i n}^{2} α \to 100 t^{99} d t = 2 s i n α c o s α d α

d t = \frac{2 s i n α c o s α d α}{100 {({s i n}^{2} α)}^{\frac{1}{100}}} = \frac{1}{50} \times {(s i n α)}^{1 - \frac{1}{50}} \times c o s α d α

\frac{\int_{0}^{\frac{π}{2}} {(1 - {s i n}^{2} α)}^{201} \times \frac{1}{50} \times {(s i n α)}^{1 - \frac{1}{50}} \times c o s α d α}{\int_{0}^{\frac{π}{2}} {(1 - {s i n}^{2} α)}^{202} \times \frac{1}{50} \times {(s i n α)}^{1 - \frac{1}{50}} \times c o s α d α} \frac{}{}

\frac{\int^{\frac{π}{2}} {c o s}^{402} α) {(s i n α)}^{1 - \frac{1}{50}} \times c o s α d α}{\int_{0}^{\frac{π}{2}} {c o s}^{404} \times {s i n}^{1 - \frac{1}{50}} \times c o s α d α}

\frac{\int_{0}^{\frac{π}{2}} {(s i n α)}^{1 - \frac{1}{50}} \times {(c o s α)}^{403} d α}{\int_{0}^{\frac{π}{2}} {(s i n α)}^{1 - \frac{1}{50}} \times {(c o s α)}^{405} d α}

n o w f o r m u l A

2 \int_{0}^{\frac{π}{2}} {s i n}^{2 p - 1} α {c o s}^{2 q - 1} α d α = \frac{⌈ (p) ⌈ (q)}{⌈ (p + q)}

2 p - 1 = 1 - \frac{1}{50} \to 2 p = 2 - \frac{1}{50} \to p = 1 - \frac{1}{100}

2 q - 1 = 403 \to 2 q = 404 \to q = 202

2 q_{1} - 1 = 405 \to 2 q_{1} = 406 \to q_{1} = 203

q_{1} = q + 1 \to ⌈ (q_{1}) = ⌈ (q + 1) \to q ⌈ (q)

p + q_{1} = p + q + 1 \to ⌈ (p + q_{1}) = ⌈ (p + q + 1) \to (p + q) ⌈ (p + q)

\frac{2 \int^{\frac{π}{2}} {s i n}^{2 p - 1} α {c o s}^{2 q - 1} α d α}{2 \int_{0}^{\frac{π}{2}} {s i n}^{2 p - 1} α {c o s}^{2 q_{1} - 1} α d α} = \frac{\frac{⌈ (p) ⌈ (q)}{⌈ (p + q)}}{\frac{⌈ (p) ⌈ (q_{1})}{⌈ (p + q_{1})}}

= \frac{⌈ (p) ⌈ (q) ⌈ (p + q_{1})}{⌈ (p) ⌈ (q_{1}) ⌈ (p + q)}

= \frac{⌈ (q) \times (p + q) ⌈ (p + q)}{q ⌈ (q) \times ⌈ (p + q)}

= \frac{p + q}{q} = \frac{1 - \frac{1}{100} + 202}{202} = \frac{100 - 1 + 20200}{20200} = \frac{20300 - 1}{20200}

= \frac{20299}{20200} R a h u l p l s c h e c k \dots