Integral with sinx in the denominator

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In summary, an integral with sinx in the denominator is a mathematical expression in the form of ∫sinx, where sinx is in the denominator. The purpose of using sinx in the denominator is to solve for the area under a curve that has a function of sinx in it. To solve this type of integral, we can use the substitution method or integration by parts. There are some special cases, such as involving multiple periods of the sine function, where we need to break up the integral and apply trigonometric identities. Real-world applications of integrals with sinx in the denominator include calculating work, finding the center of mass, and determining the area under a sine wave.
  • #1
tmt1
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I have this integral

$$\int_{}^{} \frac{x}{sinx} dx$$

How can I approach solving this? I tried u-substitution with $u = sinx$. And I get

$$\int_{}^{} \frac{x}{cosx u} du$$, so I don't think that is an option. I'm also unsure of how I would apply integration by parts. Or can I simplify the expression in some way to make this easier?
 
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  • #2
That particular integrand does not have an anti-derivative expressible in elementary terms. :)
 
  • #3
Regardless of the function not having an elementary anti-derivative, your substitution is wrong.
 

FAQ: Integral with sinx in the denominator

What does "integral with sinx in the denominator" mean?

When we talk about an integral with sinx in the denominator, we are referring to a mathematical expression in the form of ∫sinx, where sinx is in the denominator of the expression.

What is the purpose of using sinx in the denominator of an integral?

The use of sinx in the denominator of an integral allows us to solve for the area under a curve that has a function of sinx in it. This is useful in many applications, such as finding the displacement of an oscillating object or the force of a spring.

How do you solve an integral with sinx in the denominator?

To solve an integral with sinx in the denominator, we can use the substitution method or integration by parts. We first substitute u = sinx and then integrate using the appropriate method.

Are there any special cases when solving an integral with sinx in the denominator?

Yes, there are some special cases when solving an integral with sinx in the denominator. One such case is when the integral has limits of integration that involve multiple periods of the sine function. In this case, we need to break up the integral into smaller parts and apply the appropriate trigonometric identities to solve it.

What are some real-world applications of integrals with sinx in the denominator?

Integrals with sinx in the denominator have many real-world applications. Some examples include calculating the work done by a variable force, finding the center of mass of a curved object, and determining the area under a sine wave in electrical engineering.

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