How to integrate ∫ 1 y s ( 1 − y ) d y?

You can easily define your function in terms of the incomplete beta function which can be defined in terms of the hypergeometric function: B ( z ; a , b ) = ∫ 0 z x a − 1 ( 1 − x ) b − 1 d xalso note the Chebyshev integral is defined as: Ch ⁡ ( a , b ) = ∫ x p ( 1 − x ) q d x = B ( x ; 1 + p , 1 + q )in your case you get: ∫ y − s ( 1 − y ) − 1 d y = B ( y ; 1 − s , 0 )now using the relationship: B ( z ; a , b ) = z a a 2 F 1 ( a , 1 − b ; a + 1 ; z )you can get your integral in terms of the hypergeometric function, which can be accurately calculated by most softwares

Solved: "How to integrate ∫ 1 y s (..."

High School
Calculus and Analysis
Calculus 1
Integrals

Brooke Kramer

Answered question

2022-05-21

How to integrate

\int \frac{1}{y^{s} (1 - y)} d y

Answer & Explanation

Ada Harrington

Beginner2022-05-22Added 12 answers

You can easily define your function in terms of the incomplete beta function which can be defined in terms of the hypergeometric function:

B (z; a, b) = \int_{0}^{z} x^{a - 1} (1 - x)^{b - 1} d x

also note the Chebyshev integral is defined as:

Ch (a, b) = \int x^{p} (1 - x)^{q} d x = B (x; 1 + p, 1 + q)

in your case you get:

\int y^{- s} (1 - y)^{- 1} d y = B (y; 1 - s, 0)

now using the relationship:

B (z; a, b) = \frac{z^{a}}{a}_{2} F_{1} (a, 1 - b; a + 1; z)

you can get your integral in terms of the hypergeometric function, which can be accurately calculated by most softwares

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