Value of ratio of $2$ definite Integration
Evaluation of $$ 2^{2010}frac{int^{1}_{0}x^{1004}(1-x)^{1004}dx}{int^{1}_{0}x^{1004}(1-x^{2010})^{1004}dx}$$
Try: Let $displaystyle I =int^{1}_{0}x^{1004}(1-x)^{1004}dx$ and $displaystyle J =int^{1}_{0}x^{1004}(1-x^{2010})^{1004}dx$
put $x^{1005}=t$ and $displaystyle x^{1004}dx=frac{1}{1005}dt$
So $displaystyle J =frac{1}{1005}int^{1}_{0}(1-t^2)dt$
Now i am struck here , I did not understand how to solve it
Could some help me to solve it, Thanks
definite-integrals
asked Dec 11 '18 at 15:02
D Tiwari
5,4132630
add a comment |
Evaluation of $$ 2^{2010}frac{int^{1}_{0}x^{1004}(1-x)^{1004}dx}{int^{1}_{0}x^{1004}(1-x^{2010})^{1004}dx}$$
Try: Let $displaystyle I =int^{1}_{0}x^{1004}(1-x)^{1004}dx$ and $displaystyle J =int^{1}_{0}x^{1004}(1-x^{2010})^{1004}dx$
put $x^{1005}=t$ and $displaystyle x^{1004}dx=frac{1}{1005}dt$
So $displaystyle J =frac{1}{1005}int^{1}_{0}(1-t^2)dt$
Now i am struck here , I did not understand how to solve it
Could some help me to solve it, Thanks
definite-integrals
asked Dec 11 '18 at 15:02
D Tiwari
5,4132630
2
Bhai beta function ki property check kar le wikipedia se. Best of luck.
– NewBornMATH
Dec 11 '18 at 15:24
bhai tu sahi bola ye beta baba hee hain
– deleteprofile
Dec 11 '18 at 15:41
add a comment |
Evaluation of $$ 2^{2010}frac{int^{1}_{0}x^{1004}(1-x)^{1004}dx}{int^{1}_{0}x^{1004}(1-x^{2010})^{1004}dx}$$
Try: Let $displaystyle I =int^{1}_{0}x^{1004}(1-x)^{1004}dx$ and $displaystyle J =int^{1}_{0}x^{1004}(1-x^{2010})^{1004}dx$
put $x^{1005}=t$ and $displaystyle x^{1004}dx=frac{1}{1005}dt$
So $displaystyle J =frac{1}{1005}int^{1}_{0}(1-t^2)dt$
Now i am struck here , I did not understand how to solve it
Could some help me to solve it, Thanks
definite-integrals
asked Dec 11 '18 at 15:02
D Tiwari
5,4132630
Evaluation of $$ 2^{2010}frac{int^{1}_{0}x^{1004}(1-x)^{1004}dx}{int^{1}_{0}x^{1004}(1-x^{2010})^{1004}dx}$$
Try: Let $displaystyle I =int^{1}_{0}x^{1004}(1-x)^{1004}dx$ and $displaystyle J =int^{1}_{0}x^{1004}(1-x^{2010})^{1004}dx$
put $x^{1005}=t$ and $displaystyle x^{1004}dx=frac{1}{1005}dt$
So $displaystyle J =frac{1}{1005}int^{1}_{0}(1-t^2)dt$
Now i am struck here , I did not understand how to solve it
Could some help me to solve it, Thanks
definite-integrals
definite-integrals
asked Dec 11 '18 at 15:02
D Tiwari
5,4132630
asked Dec 11 '18 at 15:02
D Tiwari
5,4132630
asked Dec 11 '18 at 15:02
D Tiwari
5,4132630
asked Dec 11 '18 at 15:02
D Tiwari
5,4132630
asked Dec 11 '18 at 15:02
D Tiwari
5,4132630
5,4132630
2
Bhai beta function ki property check kar le wikipedia se. Best of luck.
– NewBornMATH
Dec 11 '18 at 15:24
bhai tu sahi bola ye beta baba hee hain
– deleteprofile
Dec 11 '18 at 15:41
add a comment |
2
Bhai beta function ki property check kar le wikipedia se. Best of luck.
– NewBornMATH
Dec 11 '18 at 15:24
bhai tu sahi bola ye beta baba hee hain
– deleteprofile
Dec 11 '18 at 15:41
2
2
Bhai beta function ki property check kar le wikipedia se. Best of luck.
– NewBornMATH
Dec 11 '18 at 15:24
Bhai beta function ki property check kar le wikipedia se. Best of luck.
– NewBornMATH
Dec 11 '18 at 15:24
bhai tu sahi bola ye beta baba hee hain
– deleteprofile
Dec 11 '18 at 15:41
bhai tu sahi bola ye beta baba hee hain
– deleteprofile
Dec 11 '18 at 15:41
add a comment |
1 Answer
1
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oldest
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With $a = 1004$ and $b = 2010$ and $B$ the beta function,
$$eqalign{I &= {frac {a{rm B} left(a,aright)}{4,a+2}}cr
J &= frac{a{rm B} left(a,frac{a+1}{b}right)}{ab + a + 1}
}
$$
and thus $$frac{I}{J} = frac{1+a+ab}{4a+2} frac{{rm B}(a,a)}{{rm B}(a,(a+1)/b)}$$
In your case I get
$$ 2^{2010} I/J = 4020$$
answered Dec 11 '18 at 15:15
Robert Israel
318k23208457
add a comment |
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1 Answer
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1 Answer
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votes
With $a = 1004$ and $b = 2010$ and $B$ the beta function,
$$eqalign{I &= {frac {a{rm B} left(a,aright)}{4,a+2}}cr
J &= frac{a{rm B} left(a,frac{a+1}{b}right)}{ab + a + 1}
}
$$
and thus $$frac{I}{J} = frac{1+a+ab}{4a+2} frac{{rm B}(a,a)}{{rm B}(a,(a+1)/b)}$$
In your case I get
$$ 2^{2010} I/J = 4020$$
answered Dec 11 '18 at 15:15
Robert Israel
318k23208457
add a comment |
With $a = 1004$ and $b = 2010$ and $B$ the beta function,
$$eqalign{I &= {frac {a{rm B} left(a,aright)}{4,a+2}}cr
J &= frac{a{rm B} left(a,frac{a+1}{b}right)}{ab + a + 1}
}
$$
and thus $$frac{I}{J} = frac{1+a+ab}{4a+2} frac{{rm B}(a,a)}{{rm B}(a,(a+1)/b)}$$
In your case I get
$$ 2^{2010} I/J = 4020$$
answered Dec 11 '18 at 15:15
Robert Israel
318k23208457
add a comment |
With $a = 1004$ and $b = 2010$ and $B$ the beta function,
$$eqalign{I &= {frac {a{rm B} left(a,aright)}{4,a+2}}cr
J &= frac{a{rm B} left(a,frac{a+1}{b}right)}{ab + a + 1}
}
$$
and thus $$frac{I}{J} = frac{1+a+ab}{4a+2} frac{{rm B}(a,a)}{{rm B}(a,(a+1)/b)}$$
In your case I get
$$ 2^{2010} I/J = 4020$$
answered Dec 11 '18 at 15:15
Robert Israel
318k23208457
With $a = 1004$ and $b = 2010$ and $B$ the beta function,
$$eqalign{I &= {frac {a{rm B} left(a,aright)}{4,a+2}}cr
J &= frac{a{rm B} left(a,frac{a+1}{b}right)}{ab + a + 1}
}
$$
and thus $$frac{I}{J} = frac{1+a+ab}{4a+2} frac{{rm B}(a,a)}{{rm B}(a,(a+1)/b)}$$
In your case I get
$$ 2^{2010} I/J = 4020$$
answered Dec 11 '18 at 15:15
Robert Israel
318k23208457
answered Dec 11 '18 at 15:15
Robert Israel
318k23208457
answered Dec 11 '18 at 15:15
Robert Israel
318k23208457
answered Dec 11 '18 at 15:15
Robert Israel
318k23208457
318k23208457
add a comment |
add a comment |
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2
Bhai beta function ki property check kar le wikipedia se. Best of luck.
– NewBornMATH
Dec 11 '18 at 15:24
bhai tu sahi bola ye beta baba hee hain
– deleteprofile
Dec 11 '18 at 15:41