How Can Integral Substitutions Simplify \(\int \frac{dx}{x^{2} e^{-2/x}}\)?

Thread starter Kuno
Start date Jun 23, 2007
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Integral

In summary, an integral substitution is a method used in calculus to simplify and solve integrals by replacing the variable of integration with a new variable. It is typically used when the integrand contains compositions of functions, and the choice of substitution depends on the form of the integrand. Not all integrals can be solved using an integral substitution, and it is important to have a variety of integration techniques. Tips for using integral substitutions include checking the answer by differentiating and practicing different types of substitutions.

Jun 23, 2007

Kuno

Homework Statement

[tex]\int \frac{dx}{x^{2} e^{\frac{-2}{x}}}[/tex]

The Attempt at a Solution

I'm not sure where to begin.

Last edited: Jun 23, 2007

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Jun 23, 2007

neutrino

Have you tried any substitutions?

Jun 23, 2007

FrogPad

What happens if you make a substitution, say: [itex] u = =\frac{2}{x} [/itex] ?

Jun 23, 2007

Kuno

I got it with u = -2/x, thanks. I'm not sure why I didn't do that in the first place.

FAQ: How Can Integral Substitutions Simplify \(\int \frac{dx}{x^{2} e^{-2/x}}\)?

What is an integral substitution?

An integral substitution is a method used in calculus to simplify and solve integrals. It involves replacing the variable of integration with a new variable in order to transform the integral into a simpler form that can be easily evaluated.

When should I use an integral substitution?

An integral substitution is typically used when the integrand (the function being integrated) contains a composition of functions, such as a polynomial inside a trigonometric function. It can also be used to solve integrals involving rational functions or exponential functions.

How do I choose the appropriate substitution?

The choice of substitution depends on the form of the integrand. In general, you want to choose a substitution that will simplify the integral and eliminate any complicated terms. Some common substitutions include u-substitution, trigonometric substitutions, and exponential substitutions.

Can I always use an integral substitution to solve an integral?

No, not all integrals can be solved using an integral substitution. Some integrals may require other techniques, such as integration by parts or partial fractions. It is important to have a variety of integration techniques in your toolkit in order to solve different types of integrals.

Are there any tips for using integral substitutions effectively?

One tip for using integral substitutions is to always check your answer by differentiating the result. This will help ensure that you have chosen the correct substitution and have solved the integral correctly. Additionally, practice and familiarity with different types of substitutions will make the process easier and more efficient.