Evaluating an indefinite integral

In summary, the indefinite integral can be evaluated by simplifying the first term to x and then integrating each term separately, resulting in an anti-derivative of \frac{x^{2}}{2} + C.
  • #1
Wm_Davies
51
0

Homework Statement


Evaluate the indefinite integral.

[tex]\int \left({\sqrt[5]{x^5}}-\frac{6}{5 x}+\frac{1}{4 x^{7}} \right) dx[/tex]


The Attempt at a Solution



O.k. the only anti-derivative I am having trouble getting is the first one [tex]{\sqrt[5]{x^5}}[/tex].

I am not sure what formula I would use or how to do it. I looked through the book, but I didn't see anything addressing this. Any help would be appreciated.

I imagine that it would be easier to write it as [tex](x^{5})^{\frac{1}{5}}[/tex]

Working through it I get [tex](\frac{x^{6}}{6})^{\frac{1}{5}}[/tex]

then I am stuck...
 
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  • #2
Remember that when you raise a power to a power, that by the Law of Exponents the powers multiply. :wink:
 
  • #3
Simplify first!
[tex]\sqrt[5]{x^5}~=~x[/tex]
 
  • #4
Mark44 said:
Simplify first!
[tex]\sqrt[5]{x^5}~=~x[/tex]

Yeah, I can't believe I missed that. So then the anti-derivative should be [tex]\frac{x^{2}}{2}[/tex] + a constant?
 
  • #5
Yes, but you won't need a constant for each term in the integrand - just one for all three.
 

1. What is an indefinite integral?

An indefinite integral is a mathematical concept that represents the anti-derivative of a function. It is used to find the original function when only its derivative is known, and is denoted by the symbol ∫ (integral sign).

2. How do you evaluate an indefinite integral?

To evaluate an indefinite integral, you use the reverse process of differentiation. This involves finding the anti-derivative of the given function and adding a constant of integration. The resulting equation is the evaluated indefinite integral.

3. What are the different methods for evaluating an indefinite integral?

There are several methods for evaluating indefinite integrals, including substitution, integration by parts, partial fraction decomposition, and trigonometric substitution. Each method is used depending on the complexity of the function being integrated.

4. What is the importance of evaluating an indefinite integral?

Evaluating indefinite integrals is important in mathematics, science, and engineering. It allows us to solve a wide range of problems involving rates of change, areas, volumes, and other physical quantities. It is also the basis for techniques used in more advanced mathematical concepts.

5. What are some common mistakes when evaluating an indefinite integral?

Some common mistakes when evaluating indefinite integrals include forgetting to add the constant of integration, incorrectly applying integration rules, and making algebraic errors. It is important to carefully check each step of the process and verify the solution to avoid these mistakes.

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