   Chapter 4.R, Problem 53E

Chapter
Section
Textbook Problem

# If f is a continuous function such that ∫ 0 x f ( t )   d t = x   sin x + ∫ 0 x f ( t ) 1 + t 2 d t for all x, find an explicit formula for, find an explicit formula for f ( x ) .

To determine

To find:

An explicit formula for fx

Explanation

1) Concept:

TheFirst FundamentalTheorem 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)

2) Calculation:

The given integral is

0xftdt=xsinx+0xf(t)1+t2dt

Now by first fundamental theorem of calculus, the given function f  is continuous

Therefore, take derivative of both sides with respect to x

ddx0xftdt=ddxxsinx+ddx0xf(t)1+t2dt

Here all the terms are functions of x

Therefore, the equation

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