How can the root integral be simplified to a more manageable form?

In summary, a root integral is a mathematical concept used to find the antiderivative of a function. It is solved by using the reverse process of differentiation and is useful in finding the original function when only its derivative is known. The main difference between a root integral and a definite integral is that a root integral results in a function, while a definite integral results in a numerical value. There are various techniques for solving root integrals, such as substitution, integration by parts, and trigonometric substitution.
  • #1
transgalactic
1,395
0
[tex]
\int \frac{dx}{x(1+2\sqrt{x}+\sqrt[3]{x})}=\int \frac{dx}{x(\sqrt{x}+1+\sqrt{x}+\sqrt[3]{x})}=
\int \frac{dx}{x(\sqrt{x}+\frac{(1-x)}{1-\sqrt[3]{x}})}
[/tex]
i got read of one root but instead i got another one
??
 
Last edited:
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  • #2
Notice that the denominator can be written as follows:

[tex]x\left(1+2x^{1/2}+x^{1/3}\right)[/tex]

[tex]x\left(1+2x^{3/6}+x^{2/6}\right)[/tex]

Let [itex]u=x^{1/6}[/itex].
 

Related to How can the root integral be simplified to a more manageable form?

1. What is a root integral?

A root integral is a mathematical concept used to find the antiderivative (or integral) of a function. It is also known as the inverse function of differentiation.

2. How do you solve a root integral?

To solve a root integral, you need to use the reverse process of differentiation. This involves finding the antiderivative of the given function and adding a constant of integration.

3. What is the purpose of a root integral?

The purpose of a root integral is to find the original function when only its derivative is known. This is useful in many fields of science, such as physics and engineering, where the rate of change of a variable is known but the original function is needed.

4. What is the difference between a root integral and a definite integral?

The main difference between a root integral and a definite integral is that a root integral finds the antiderivative of a function, while a definite integral calculates the area under a curve between two specific points. A root integral typically results in a function, while a definite integral results in a numerical value.

5. Are there any special techniques for solving root integrals?

Yes, there are various techniques for solving root integrals, such as substitution, integration by parts, and trigonometric substitution. It is important to know and understand these techniques in order to solve more complex root integrals.

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