   Chapter 4.R, Problem 54E

Chapter
Section
Textbook Problem

# Find a function f and a value of the constant a such that 2 ∫ a x f ( t )   d t = 2 ​   sin x − 1

To determine

To find:

A function fx and the value of the constant a

Explanation

1) Concept:

i) The First Fundamental Theorem for calculus: If f is continuous on [a, b], then the function g is defined by g(x)=abftdtaxb  is continuous on [a,b] and differentiable on (a,b) and g(x)=f(x)

ii) The Second Fundamental Theorem for calculus: If f is continuous on [a, b], then the function is defined by abftdt=Fb-Fa𝑤here F  is an antiderivative of f

2) Given:

2axftdt=2sinx-1

3) Calculation:

The given integral is

2axftdt=2sinx-1

axftdt=2sinx-12

Assuming f is continuous let us apply first fundamental theorem of calculus

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