   Chapter 5.5, Problem 91E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# If a and b are positive numbers, show that ∫ 0 1 x a ( 1 − x ) b d x = ∫ 0 1 x b ( 1 − x ) a d x

To determine

To prove: The integral functions.

Explanation

Given:

The integral function of left hand side is 01xa(1x)bdx.

The region lies between x=0 and x=1.

Calculation:

Take the consideration as follows:

u=1xx=1u (1)

Differentiate both sides of the Equation (1).

du=dxdu=dx

Calculate the lower limit value of u using Equation (1).

Substitute 0 for x in Equation (1).

u=10=1

Calculate the upper limit value of u using Equation (1).

Substitute 1 for x in Equation (1).

u=11=0

The integral function is,

01xa(1x)bdx (2)

Apply lower and upper limits for u in Equation (2).

Substitute (1u) for x, u for (1x), and (du) for dx in Equation (2)

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Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 