To compare an integral with an identity

In summary, the conversation is discussing how to prove the inequality for a_n in terms of n and pi, and the suggested method involves using comparison and the fact that n and x can be manipulated in the integral. The conversation also mentions the importance of showing work and seeking feedback on a solution.
  • #1
Hummingbird25
86
0
Integral inequality and comparison

Homework Statement



Prove the inequality

[tex]\frac{2}{(n+1) \cdot \pi} \leq a_n \leq \frac{2}{n \pi}}[/tex]

where [tex]a_n = \int_{0}^{\pi} \frac{sin(x)}{n \cdot \pi +x} dx[/tex]

and [tex]n \geq 1[/tex]

The Attempt at a Solution



Proof:

If n increased the left side of inequality because smaller every time it increases while the integral becomes larger, then by camparison the right side of the inequality always will become larger, than the integral.

Is this proof enough?

Sincerely
Maria.
 
Last edited:
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  • #2
well try to use the fact that

[tex]n\pi\leq \pi n+x \leq n\pi+\pi[/tex], then

[tex]\frac{1}{n\pi+\pi}\leq\frac{1}{n\pi+x}\leq\frac{1}{n\pi}[/tex], multiply both sides by sinx, we get, notice that x goes only from 0 to pi, so sinx is always positive, so

[tex]\frac{sin x}{n\pi+\pi}\leq\frac{sin x}{n\pi+x}\leq\frac{sin x}{n\pi}[/tex], now

[tex]\int_0^{\pi}\frac{sin x }{n\pi+\pi}dx\leq\int_0^{\pi}\frac{sin x}{n\pi+x}dx\leq\int_0^{\pi}\frac{sin x}{n\pi}dx[/tex],
now
[tex]\frac{1}{n\pi+\pi}\int_0^{\pi}sin xdx\leq\int_0^{\pi}\frac{sin x}{n\pi+x}dx\leq\frac{1}{n\pi}\int_0^{\pi}\sin xdx[/tex],
I think this will work!
 
  • #3
I shouldn't have done the whole thing for you, since you are on the homework forum, and you are supposed to show your work, however, i hope u got it now!
 
  • #4
sutupidmath said:
I shouldn't have done the whole thing for you, since you are on the homework forum, and you are supposed to show your work, however, i hope u got it now!

I get it now. I will present something that I have been working later I hope you will review.
 
  • #5
Hummingbird25 said:
I get it now. I will present something that I have been working later I hope you will review.

well, someone will!
 

1. What is an integral?

An integral is a mathematical concept that represents the area under a curve in a graph. It is typically denoted by the symbol ∫ and is used to find the total value of a function over a given interval.

2. What is an identity in relation to integrals?

An identity is a mathematical equation that is true for all values of the variables involved. In the context of integrals, an identity can be used to simplify or manipulate the integral expression in order to solve it more easily.

3. How do you compare an integral with an identity?

To compare an integral with an identity, you need to look for similarities between the integral expression and the identity. These similarities can be used to simplify the integral or transform it into a form that can be easily solved using the identity.

4. Why is it useful to compare an integral with an identity?

Comparing an integral with an identity can make it easier to solve the integral and find its exact value. It can also help in proving the validity of the integral expression or identifying any mistakes in the calculation.

5. Are there any common identities used in integral comparisons?

Yes, there are several common identities used in integral comparisons, such as the power rule, the substitution rule, and the trigonometric identities. These identities can be applied to specific types of integrals to simplify or solve them more efficiently.

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