   Chapter 4.P, Problem 1P

Chapter
Section
Textbook Problem

# If x   sin x   π x = ∫ 0 x 2 f ( t )   d t , where f is a continuous function, find f(4).

To determine

To find:

f4

Solution:

π2

Explanation

1) Concept:

The Fundamental Theorem of Calculus

Fx=ddx0xf(t)dt  then Fx=f(x)

2) Given:

xsinπx=0x2ftdt

3) Calculation:

Here,

xsinπx=0x2ftdt

Substitute x2=u, (x=u)

u sinπu=0uftdt

Differentiate with respect to  u:

12usinπu+u cosπu·π2u=ddu0uftdt

By using Fundamental Theorem of Calculus,

12usinπu+u cosπu·π2u=fu

Tofind  f4, substitute u=4

f4=124sinπ4+4 cosπ4·π24

=14sin2π+2 cos2π·π4

Substitute sin2π=0 &  cos2π=1

=140+2 1·π4

=π2

Conclusion:

f4=π2

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