# Luc's question at Yahoo Answers regarding an indefinite integral

• MHB
• MarkFL
In summary, we are asked to find the indefinite integral of x^4(5x^5+1)^7 dx. By using the u-substitution method, we can rewrite the integral as (1/200)(5x^5+1)^8 + C. We encourage further calculus problems to be posted in the forum.
MarkFL
Gold Member
MHB
Here is the question:

Find the indefinite integral?

x^4(5x^5+1)^7 dx

Here is a link to the question:

Find the indefinite integral? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.

Re: Luc's question at Yahoo! Answers regarding an indedinte integral

Hello Luc,

$$\displaystyle \int x^4\left(5x^5+1 \right)^7\,dx$$

It we use the $u$-substitution:

$$\displaystyle u=5x^5+1\,\therefore\,du=25x^4\,dx$$

we may rewrite the integral as:

$$\displaystyle \frac{1}{25}\int u^7\,du=\frac{1}{25}\left(\frac{u^8}{8} \right)+C=\frac{1}{200}u^8+C$$

Now, back-substituting for $u$, we may state:

$$\displaystyle \int x^4\left(5x^5+1 \right)^7\,dx=\frac{1}{200}\left(5x^5+1 \right)^8+C$$

To Luc and any other guests viewing this topic, I invite and encourage you to post other calculus problems in our http://www.mathhelpboards.com/f10/ forum.

Best Regards,

Mark.

## 1. What is an indefinite integral?

An indefinite integral is a mathematical concept that represents the antiderivative of a function. It is the inverse operation of differentiation, and it is used to calculate the original function when given its derivative.

## 2. How do you solve an indefinite integral?

To solve an indefinite integral, you must use the fundamental theorem of calculus, which states that the antiderivative of a function can be found by integrating the function with respect to its variable and adding a constant term. This can be done using integration techniques such as substitution, integration by parts, or using tables of integrals.

## 3. What is the purpose of finding an indefinite integral?

The purpose of finding an indefinite integral is to determine the original function when given its derivative. This is useful in various fields of science and engineering, such as physics, economics, and engineering, where the rate of change of a variable is known, but the original function is unknown.

## 4. Can an indefinite integral have multiple solutions?

Yes, an indefinite integral can have multiple solutions, as the constant term that is added in the integration process can take on any value. This means that there can be many different functions that have the same derivative, making it important to include the constant term in the solution.

## 5. Is there a difference between an indefinite integral and a definite integral?

Yes, there is a difference between an indefinite integral and a definite integral. A definite integral has specific limits of integration and gives a numerical value, while an indefinite integral does not have limits of integration and represents a family of functions that differ by a constant term.

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