   Chapter 10.3, Problem 77E

Chapter
Section
Textbook Problem

# Integration by Substitution Use integration by substitution to show that if y is a continuous function of x on the interval a ≤ x ≤ b , where x = f ( t ) and x = g ( t ) . then ∫ a b y   d x = ∫ 2 t g ( t ) f ' ( t )   d t where f ( t 1 ) = a , f ( t 2 ) = b , and both g and f ' are continuous on [ t 1 , t 2 ] .

To determine

To prove: The equation abydx=t1t2g(t)f(t)dt when y=g(t) and x=f(t), axb and f(t1)=a,f(t2)=b both the functions g and f are continuous on [t1,t2].

Explanation

Given:

The parametric function, x=f(t) and y=g(t) and also, f(t1)=a and f(t2)=b. Both g and f are continuous on [t1,t2].

Proof:

The function x=f(t) is a parametric function.

The function y is continuous function on the interval axb. Given, the value of function x=f(t) at t1 is,

f(t1)=a

The value of function x=f(t) at t2 is,

f(t2)=b

Differentiate the function x=f(t) with respect to t

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