Integrating Misc Integral with \int \frac{x}{\sqrt{3-x^4}}dx

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In summary, the purpose of integrating Misc Integral with <code>\int \frac{x}{\sqrt{3-x^4}}dx</code> is to find the antiderivative and evaluate the definite integral, which has practical applications in physics, engineering, and economics. To solve this integral, we can use integration principles and look for simplifications. While calculators and computer programs can also solve this integral, it is important to understand the manual steps. Special cases and restrictions, such as the domain of the function, need to be considered when integrating this expression. Real-life applications include calculating work, center of mass, and consumer/producer surplus.
  • #1
nameVoid
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[tex]
\int \frac{x}{\sqrt{3-x^4}}dx
[/tex]
[tex]
u=x^2
[/tex]
[tex]
du=2xdx
[/tex]
[tex]
\frac{1}{2}\int\frac{du}{\sqrt{3-u^2}}
[/tex]
[tex]
u=\sqrt{3}sinT
[/tex]
[tex]
du=\sqrt{3}cosTdT
[/tex]
[tex]
\frac{1}{2}\int \frac{\sqrt{3}cosT}{\sqrt{3}cosT}dT

[/tex]
[tex]
\frac{x}{2}+C
[/tex]
 
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  • #2
Not x/2+C. T/2+C.
 
  • #3
You need not do the second u-substitution. Use an integral table to solve it once you have done the first u-substitution. You can google 'integral table' to find one if you don't have one in your book.
 

1. What is the purpose of integrating Misc Integral with \int \frac{x}{\sqrt{3-x^4}}dx?

The purpose of integrating Misc Integral with \int \frac{x}{\sqrt{3-x^4}}dx is to find the antiderivative, or the original function, of the given expression. This allows us to calculate the definite integral and find the area under the curve, which has many practical applications in physics, engineering, and economics.

2. How do you approach solving this integral?

To solve this integral, we need to use the principles of integration, such as the power rule, substitution, or integration by parts. We also need to identify any patterns or simplifications that can be made to the expression to make it easier to integrate.

3. Can we use a calculator or computer program to solve this integral?

Yes, there are many calculators and computer programs that can solve integrals, including the one involving \int \frac{x}{\sqrt{3-x^4}}dx. However, it is important to understand the principles and steps involved in solving integrals by hand, as it can help with understanding and verifying the results.

4. Are there any special cases or restrictions when integrating this expression?

Yes, when integrating \int \frac{x}{\sqrt{3-x^4}}dx, we need to be careful of the domain of the function. In this case, the expression is undefined when x=0 or when 3-x^4<0, so we need to take into account these restrictions when evaluating the integral.

5. What are some real-life applications of integrating this expression?

Integrating \int \frac{x}{\sqrt{3-x^4}}dx has many real-life applications, such as calculating the work done by a variable force or finding the center of mass of an object with varying density. It is also used in economics to calculate consumer surplus and producer surplus in a market with changing prices.

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