Solving Integral Problem: Limit of Sum of Cosines

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In summary, the conversation discusses an integral problem involving a limit inside the integrand. It is discovered that the limit can be written as an integral and can be computed to be equal to sin(x)/x. Integrating from 0 to infinity results in pi/2. The person is unsure how to compute the integral inside and is seeking guidance.
  • #1
Upiór
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I have a problem with this integral. By the way, it's not my homework:)

[tex]\int\limits_{0}^{+\infty}(\lim\limits_{n\rightarrow+\infty} \displaystyle\frac{1+\cos\frac{x}{n}+\cos\frac{2x}{n}+\ldots+\cos \frac{(n-1)x}{n}}{n} \right))dx[/tex]
 
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  • #2
The limit inside the integrand can be written as an integral. If you compute that integral, you find that it is equal to sin(x)/x. If you integrate that from zero to infinity, you get pi/2.
 
  • #3
Hm, but how to compute this integral inside? Why is it equal to sinx/x? I cannot see the way...

Step by step, please:)
 

FAQ: Solving Integral Problem: Limit of Sum of Cosines

What is an integral?

An integral is a mathematical concept that represents the area under a curve on a graph. It is used to find the total value of a function within a specific interval.

How do you solve an integral problem?

To solve an integral problem, you must first determine the limits of integration, which are the starting and ending points of the interval. Then, you can use various techniques such as substitution, integration by parts, or trigonometric identities to evaluate the integral.

What is the "limit of sum of cosines"?

The limit of sum of cosines refers to the value that a series of cosine terms approaches as the number of terms increases towards infinity. It is often used in integral problems involving trigonometric functions.

Why is the "limit of sum of cosines" important?

The "limit of sum of cosines" allows us to evaluate integrals involving trigonometric functions that cannot be solved using traditional techniques. It also has applications in physics, engineering, and other sciences where trigonometry is commonly used.

Are there any tips for solving integral problems involving the "limit of sum of cosines"?

One helpful tip is to use trigonometric identities to simplify the integral before taking the limit of sum of cosines. It is also important to pay attention to the limits of integration and make sure they are properly defined. Practice and familiarity with different techniques for solving integrals can also improve your ability to solve problems involving the limit of sum of cosines.

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