Integrating Absolute Value in Trigonometric Integrals

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In summary, integrating 1/sin(x) involves finding the antiderivative of the function, which is important in various fields such as physics and engineering. The integral can be solved using different methods depending on the form of the function. There are special cases, such as when the integration limits include sin(x) = 0, and the integral can have real-life applications, such as calculating work or determining probabilities.
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NanakiXIII
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Homework Statement



I need to solve the integral

[tex]\int \frac{dx}{\sin{x} |\sin{x}|}.[/tex]

Homework Equations



-

The Attempt at a Solution



I know it's possible, the solution is [tex]-\frac{\cos{x} |\sin{x}|}{\sin^2{x}}[/tex], and I probably need to write the absolute as the square root of a square. I also probably need a substitution to simplify. I'm not having any luck finding such a substitution, however.
 
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  • #2
Integrate it for sinx>0, then for sinx<0. Compare.
 
  • #3
That worked perfectly. Thanks. I suppose that's the usual way to tackle an integral over an absolute value?
 

FAQ: Integrating Absolute Value in Trigonometric Integrals

What does it mean to integrate 1/sinx?

Integrating 1/sin(x) is the process of finding the antiderivative of the function 1/sin(x). This means finding a function whose derivative is equal to 1/sin(x).

Why is it important to integrate 1/sinx?

Integrating 1/sin(x) is important in many areas of science and mathematics, such as physics, engineering, and statistics. It allows us to find the total area under the curve of the function, which can help us solve real-world problems and make predictions.

How do you integrate 1/sinx?

The integral of 1/sin(x) can be solved using various techniques, such as substitution, integration by parts, or trigonometric identities. The specific method used will depend on the form of the function and the knowledge and preferences of the person solving it.

Are there any special cases when integrating 1/sinx?

Yes, there are a few special cases when integrating 1/sin(x). One example is when the integration limits include the point where sin(x) = 0. In this case, the integral is undefined. Another special case is when the function is in the form of 1/sin^2(x), which can be solved using a trigonometric substitution.

How can integrating 1/sinx be applied in real-life situations?

Integrating 1/sin(x) can be applied in many real-life situations, such as calculating the work done by a force, finding the average value of a periodic function, or determining the probability of an event occurring. It is a powerful tool in solving problems involving curves and periodic functions.

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