Integrating the Sine Integral: Solving the Challenging Integral of sinx/x

In summary, the problem is to integrate the function sinx/x and the method recommended is to use the Taylor series approximation for the sine integral function rather than integration by parts.
  • #1
Roni1985
201
0
1. The problem statement, all variables and given/known

Homework Statement



[tex]\int \frac{sinx}{x}dx[/tex]

Homework Equations


The Attempt at a Solution



Which method should work here? I tried integration by parts and it looks too much.
Is there a way to solve it without approximating it with the Taylor expansion of sinx ?

Thanks
 
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  • #2


That's the 'sine integral'. It's defined as a special function Si(x), you can't express it terms of a simple form using powers of x and trig functions. Approximating by taylor series is the way to go.
 
  • #3


Dick said:
That's the 'sine integral'. It's defined as a special function, you can't express it terms of a simple form using powers of x and trig functions. Approximating by taylor series is the way to go.

I see. Thanks very much for the explanation.
 

1. What is integration by parts?

Integration by parts is a mathematical technique used to solve integrals that involve the product of two functions. It involves breaking down the integral into two parts and using the product rule of differentiation to simplify the equation.

2. When should I use integration by parts?

Integration by parts is useful when the integral involves a product of two functions, and one of the functions is easier to integrate than the other. In such cases, using integration by parts can simplify the integral and make it easier to solve.

3. How do I choose which function to integrate and which to differentiate in integration by parts?

The general rule is to choose the function that becomes simpler after being differentiated as the one to differentiate, and the function that becomes simpler after being integrated as the one to integrate. This may require some trial and error, but with practice, it becomes easier to determine which function to choose.

4. Is there a specific order to follow when using integration by parts?

Yes, there is a specific order to follow when using integration by parts, known as the LIATE rule. LIATE stands for Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, and Exponential functions. The idea is to choose the function that is lower on this list as the one to integrate, as it is likely to become simpler after being differentiated.

5. Can integration by parts be used for definite integrals?

Yes, integration by parts can be used for both indefinite and definite integrals. When using it for definite integrals, the limits of integration need to be substituted into the final solution to get the correct answer.

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