Integrating sinx/x: Solving a Common Homework Question

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In summary, the integral of sin(x)/x can be solved using the sine integral function Si(x) and cannot be expressed in terms of elementary functions. This is a proven theorem and not just a matter of not being able to find the right approach.
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elabed haidar
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Homework Statement



simple question what is the integral of sinx/x i really tried but i couldn't find the trick

Homework Equations





The Attempt at a Solution

 
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  • #2
There is a very simple "trick":
[tex]\int\frac{sin(x)}{x}dx= Si(x)+ C[/tex]
because "Si(x)", the "sine integral
http://mathworld.wolfram.com/SineIntegral.html
is defined to be that integral.

sin(x)/x cannot be integrated in terms of elementary functions.
 
  • #3
To expand a bit on the answer given by HallsofIvy: the indefinite integral of f(x) = sin(x)/x cannot be done in finitely many terms involving elementary functions, such as powers, roots, trig functions, exponentials, logarithms, etc. This is a rigorously proven *theorem*. It means that even if you allow yourself 100 billion pages you could not write the answer "explicitly" on that amount of paper, using only elementary functions. Of course, you could immediately write an infinite series answer, but we are not counting that as "explicit". Again, this is a *theorem*; it is not just that nobody has been smart enough to know how to do it, but rather, that it is impossible to do.

RGV
 
Last edited:
  • #4
thank you very much so that was the trick people are talking about lol thank you all but thank you ray vickson for the explanation
 

1. What is the formula for integrating sinx/x?

The formula for integrating sinx/x is ∫sindx = -cosx + C. This is known as the antiderivative of sinx, which is the function that when differentiated gives sinx.

2. Why is integrating sinx/x a common homework question?

Integrating sinx/x is a common homework question because it involves using one of the fundamental integration formulas and requires applying certain techniques and rules, making it a good practice for students learning integration.

3. How do I solve an integral with sinx/x?

To solve an integral with sinx/x, you can use the formula ∫sindx = -cosx + C and integrate the function directly. However, if the integral is more complex, you may need to employ substitution or integration by parts to solve it.

4. Can I use any other methods to integrate sinx/x?

Yes, in addition to the formula ∫sindx = -cosx + C, you can also use trigonometric identities such as sin2x = 2sinxcosx or sin^2x = (1-cos2x)/2 to simplify the integral and make it easier to integrate.

5. Are there any special cases when integrating sinx/x?

Yes, there are special cases when integrating sinx/x. For example, if the integral is evaluated at 0, the result would be undefined since the function is not defined at x=0. Also, if the integral is evaluated at ∞, the result would be π/2.

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