Find Analytic Expression for Integral with Approximations

In summary, an integral is a mathematical concept used to calculate the area under a curve in a graph. An approximation, on the other hand, is a simplified version of a mathematical expression used to estimate more complex expressions. Finding analytic expressions for integrals with approximations allows for solving complex problems and providing more accurate estimates. Common methods for this include the trapezoidal rule, Simpson's rule, and Gaussian quadrature. However, there are limitations to using approximations, as their accuracy depends on factors such as the complexity of the function and the method used. In some cases, other techniques may be necessary for a more precise solution.
  • #1
venkaiah
1
0
Find the closed form (or) analytic expression form for the following integral

$$
\hspace{0.3cm} \large {\int_{0} ^{\infty} \frac{\frac{1}{x^4} \hspace{0.1cm} e^{- \frac{r}{x^2}}\hspace{0.1cm}e^{- \frac{r}{z^2}} }{ \frac{1}{x^2} \hspace{0.1cm} e^{- \frac{r}{x^2}}+ \frac{1}{y^2} \hspace{0.1cm} e^{- \frac{r}{y^2}}}} dr \hspace{.2cm} ; \hspace{1cm} x>0,y>0,z>0 $$ where $ x $ ,$ y $ and $z $ are constants and independent of $ r $.
 
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  • #2
Hi venkaiah and welcome to MHB! :D

Any thoughts on how to begin?
 

FAQ: Find Analytic Expression for Integral with Approximations

What is an integral?

An integral is a mathematical concept that represents the area under a curve in a graph, and is used to calculate the total value of a function over a given interval.

What is an approximation?

An approximation is a simplified version of a mathematical expression or function that is used to estimate the value of a more complex expression. It is often used when the exact solution is difficult to obtain.

Why do we need to find analytic expressions for integrals with approximations?

Finding analytic expressions for integrals with approximations allows us to solve mathematical problems that would otherwise be too complex to solve. These expressions can also provide a more precise estimate of the integral value compared to other approximation methods.

What are some common methods for finding analytic expressions for integrals with approximations?

The most common methods for finding analytic expressions for integrals with approximations include the trapezoidal rule, Simpson's rule, and Gaussian quadrature. These methods involve dividing the integral into smaller segments and approximating the curve within each segment.

Are there any limitations to using approximations for integrals?

Yes, there are limitations to using approximations for integrals. The accuracy of the approximation depends on the complexity of the function, the number of segments used, and the method of approximation. In some cases, these methods may not provide an accurate enough solution and other techniques may need to be used.

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