Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![If \( a \) and \( b \) are positive numbers, show that
\[
\int_0^1 x^a (1-x)^b \, dx = \int_0^1 x^b (1-x)^a \, dx
\]
Let \( u = 1 - x \). Then, \( x = \boxed{1 - t} \) and \( du = \boxed{-dx} \).
Use this substitution to rewrite the integral in terms of \( u \).
\[
\int_0^1 x^a (1-x)^b \, dx = \int_1^0 \boxed{} \, u^b (-du) = \int_0^1 \boxed{} \, u^b (1-u)^a \, du
\]
Then, replacing \( u \) with \( x \) results in the integral,
\[
\int_0^1 \boxed{} \, (1-x)^a \, dx
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3fedbef1-4110-4bb3-9906-fd5663476380%2F74a96490-5373-4370-938e-337697d9dbe9%2Fd8i7kt_processed.jpeg&w=3840&q=75)
Transcribed Image Text:If \( a \) and \( b \) are positive numbers, show that
\[
\int_0^1 x^a (1-x)^b \, dx = \int_0^1 x^b (1-x)^a \, dx
\]
Let \( u = 1 - x \). Then, \( x = \boxed{1 - t} \) and \( du = \boxed{-dx} \).
Use this substitution to rewrite the integral in terms of \( u \).
\[
\int_0^1 x^a (1-x)^b \, dx = \int_1^0 \boxed{} \, u^b (-du) = \int_0^1 \boxed{} \, u^b (1-u)^a \, du
\]
Then, replacing \( u \) with \( x \) results in the integral,
\[
\int_0^1 \boxed{} \, (1-x)^a \, dx
\]
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