Chapter 5.5, Problem 109E

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

Chapter
Section

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

# Rewriting Integrals(a) Show that ∫ 0 1 x 3 ( 1 − x ) 8 d x = ∫ 0 1 x 8 ( 1 − x ) 3 d x . (b) Show that ∫ 0 1 x a ( 1 − x ) b d x = ∫ 0 1 x b ( 1 − x ) a d x .

(a)

To determine

To prove: The integral 01x3(1x)8dx=01x8(1x)3dx.

Explanation

Given: Theintegral is:

âˆ«01x3(1âˆ’x)8dx=âˆ«01x8(1âˆ’x)3dx

Formula used: If the function, u=g(x) has a continuous derivative on the closed interval [a,b] and f is continuous on the range of g.Then,

âˆ«abf(g(x))gâ€²(x)dx=âˆ«g(a)g(b)f(u)du

The properties of definite integral are:

âˆ«abf(x)dx=âˆ«abf(u)duâˆ«abf(x)dx=âˆ’âˆ«baf(x)dx

Proof: Consider the left-hand side of the provided integral,

âˆ«01x3(1âˆ’x)8dx

Let u=1âˆ’x.Then, du=âˆ’dx.

If x=0. Then, u=1.

And if x=1. Then, u=0

(b)

To determine

To prove: The provided integral.

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