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

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Answer 1

Textbook Question

Chapter 4.P, Problem 1P

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

Expert Solution & Answer

To determine

To find:

f4

Answer to Problem 1P

Solution:

π2

Explanation of Solution

1) Concept:

The Fundamental Theorem of Calculus

Fx=ddx∫0xf(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=ddu∫0uftdt

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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Answer 2

Textbook Question

Chapter 4.P, Problem 1P

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

Expert Solution & Answer

To determine

To find:

f4

Answer to Problem 1P

Solution:

π2

Explanation of Solution

1) Concept:

The Fundamental Theorem of Calculus

Fx=ddx∫0xf(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=ddu∫0uftdt

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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Answer 3

Textbook Question

Chapter 4.P, Problem 1P

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

Expert Solution & Answer

To determine

To find:

f4

Answer to Problem 1P

Solution:

π2

Explanation of Solution

1) Concept:

The Fundamental Theorem of Calculus

Fx=ddx∫0xf(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=ddu∫0uftdt

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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Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 4

Textbook Question

Chapter 4.P, Problem 1P

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

Expert Solution & Answer

To determine

To find:

f4

Answer to Problem 1P

Solution:

π2

Explanation of Solution

1) Concept:

The Fundamental Theorem of Calculus

Fx=ddx∫0xf(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=ddu∫0uftdt

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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Answer 5

Textbook Question

Answer 6

Textbook Question

Answer 7

Expert Solution & Answer

To determine

To find:

f4

Answer to Problem 1P

Solution:

π2

Explanation of Solution

1) Concept:

The Fundamental Theorem of Calculus

Fx=ddx∫0xf(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=ddu∫0uftdt

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

Answer 8

Expert Solution & Answer

To determine

To find:

f4

Answer to Problem 1P

Solution:

π2

Explanation of Solution

1) Concept:

The Fundamental Theorem of Calculus

Fx=ddx∫0xf(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=ddu∫0uftdt

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

Answer 9

Expert Solution & Answer

Answer 10

Expert Solution & Answer

Answer 11

To determine

To find:

f4

Answer 12

Answer to Problem 1P

Solution:

π2

Answer 13

Explanation of Solution

1) Concept:

The Fundamental Theorem of Calculus

Fx=ddx∫0xf(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=ddu∫0uftdt

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

Answer 14

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

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Answer to Problem 1P

Explanation of Solution

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