Calculative Integral

Calculus Level 3

2 2010 ∫ 0 1 x 1004 ( 1 − x ) 1004 d x ∫ 0 1 x 1004 ( 1 − x 2010 ) 1004 d x = ? 2^{2010}\frac{\displaystyle \int_{0}^{1}x^{1004}(1-x)^{1004}dx}{\displaystyle \int_{0}^{1}x^{1004}(1-x^{2010})^{1004}dx} = ?

3600 1005 4020 2010 2

Aman Rajput by Aman Rajput , 25, India

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2 solutions

Parth Sankhe
Dec 2, 2018

Numerator :

∫ 0 1 ( x ( 1 − x ) ) 1004 = 2 ∫ 0 ½ ( x ( x − 1 ) ) 1004 = 2 ∫ 0 ½ ( ( ½ + 0 − x ) ( 1 − ( ½ + 0 − x ) ) ) 1004 \int ^{1}_0 (x(1-x))^{1004}=2\int ^{½}_0 (x(x-1))^{1004 }= 2 \int ^{½}_0 ((½+0-x)(1-(½+0-x)))^{1004}

= 2 ∫ 0 ½ ( ¼ − x 2 ) 1004 2\int ^{½} _0 (¼-x^2)^{1004}

Putting x = t 2 x=\frac {t}{2} , we get

= ∫ 0 1 ( 1 − t 2 ) 1004 4 1004 \int ^{1}_0 \frac {(1-t^2)^{1004}}{4^{1004}}

Denominator :

Put x 1005 = t x^{1005}=t , hence x 1004 d x = d t 1005 x^{1004}dx=\frac {dt}{1005}

= ∫ 0 1 ( 1 − t 2 ) 1004 1005 \int ^{1}_0 \frac {(1-t^2)^{1004}}{1005}

Divide the above two integrals to cancel out the common integral, multiply by 2 2010 2^{2010} , and you should get the answer as 4 × 1005 = 4020 4×1005=4020

2 Helpful 0 Interesting 0 Brilliant 0 Confused

One suggestion Parth, use \displaystyle in front of \int to make the integral more visible and use \dfrac intead of \frac to make fractions bigger in case you dont know:)

A Former Brilliant Member - 2 years, 6 months ago

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Cool! Thanks!

Parth Sankhe - 2 years, 6 months ago
Aaghaz Mahajan
Dec 2, 2018

A simple application of Beta function ............

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