How to find the integral of (3x^3 - 1)/x from 1 to e

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In summary, the process for finding the integral of (3x^3 - 1)/x from 1 to e involves using techniques such as substitution, integration by parts, or partial fraction decomposition. However, substitution may be the easiest and most efficient method in this specific case. To use substitution, you must first identify a part of the function that can be replaced by a new variable, and then integrate with respect to that variable. This process essentially finds the area under the curve of the function and is a fundamental concept in calculus. While calculators can also be used to find integrals, it is important to understand the concepts behind the process.
  • #1
IntegrateMe
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[tex]\int_{1}^{e} \frac{3x^3-1}{x} dx[/tex]

What method? I haven't seen a problem like this before.
 
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  • #2


Remember how (a+b)/c = a/c + b/c ? Use a similar principle.
 
  • #3


Ah, OK so:

[tex]\int_{1}^{e} 3x^2 -\frac{1}{x} dx[/tex]

In which case this becomes a simple integration problem.

Thank you!
 

1. What is the process for finding the integral of (3x^3 - 1)/x from 1 to e?

The process for finding the integral of a function involves using techniques such as substitution, integration by parts, or partial fraction decomposition. In this specific case, the integral can be solved using the substitution method.

2. How do I use substitution to find the integral of (3x^3 - 1)/x from 1 to e?

To use substitution, first identify a part of the function that can be replaced by a new variable. In this case, let u = 3x^3 - 1. Then, solve for x in terms of u to get x = (u + 1)^(1/3). Substitute this value into the original integral, and then integrate with respect to u. Finally, substitute back in the original variable x to get the final answer.

3. Can I use other methods besides substitution to find this integral?

Yes, you can also use integration by parts or partial fraction decomposition. However, in this specific case, substitution may be the easiest and most efficient method.

4. Can I use a calculator to find the integral of (3x^3 - 1)/x from 1 to e?

Yes, many calculators have built-in integral functions that can solve this problem for you. However, it is always important to understand the process and concepts behind finding integrals, rather than solely relying on a calculator.

5. What does it mean to find the integral of a function?

Finding the integral of a function is essentially finding the area under the curve of that function. It is a fundamental concept in calculus and is used to solve problems in various fields such as physics, engineering, and economics.

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