Stuck in Evaluating an Indefinite Integral

In summary, eumyang attempted to use substitution to solve for the integral but had an issue before. They then tried using u = x^8 and got the same result as before. They then tried using u = x^8 and got the same result as before.
  • #1
whatisphysics
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Homework Statement



Evaluate the indefinite integral.

Homework Equations


I tried to use substitution.


The Attempt at a Solution


I'm not so good with using the LaTex codes...so I attached a file I made in Word.

Thanks in advance!
 

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  • #2
Try using u = x^8
 
  • #3
iamalexalright said:
Try using u = x^8

The x^8 from e^(x^8) or x^7?
 
  • #4
I don't understand your question

But besides a poor choice of u in your original solution you don't carry out the integral properly

With my choice of u (u = x^8) compute du/dx and show me the new integral
 
  • #5
Okay, I've now ended up with

\\frac{1}{7}\\int\\frac{e^u}{x^7} du
 
  • #6
You should show me the steps you take to get there!

plus use the carrot to show superscripts with LaTeX: e^{u}

What is the derivative of u = x^8 with respect to x?
 
  • #7
Can you show me how you would do it?
 
  • #8
Alright, the integral is:

[tex]\int x^{7}e^{x^{8}}dx[/tex]

I'll choose u = x^8
[tex]du = 8x^{7}dx[/tex]

Here is where you had an issue before, notice that in the original integral I do have x^7dx in it ! This implies I don't need to divide by x^7, I just use the substitution:

[tex]\frac{1}{8}du = x^{7}dx[/tex]

Can you finish it from here?
 
  • #9
iamalexalright said:
Alright, the integral is:

[tex]\int x^{7}e^{x^{8}}dx[/tex]

I'll choose u = x^8
[tex]du = 8x^{7}dx[/tex]

Here is where you had an issue before, notice that in the original integral I do have x^7dx in it ! This implies I don't need to divide by x^7, I just use the substitution:

[tex]\frac{1}{8}du = x^{7}dx[/tex]

Can you finish it from here?

Okay, I tried what you said. Can you check it, thanks! And how do I go on from there, I'm not very good at this, but I still want to learn!
 

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  • #10
[tex]\begin{aligned}
dx &= \frac{du}{8x^7} \\
\int \frac{x^{7} \cdot e^u}{8x^{7}}dx &= ...
\end{aligned}[/tex]
This is the 2nd time I saw someone work a u-substitution this way. Why isolate dx? I think it makes things complicated. Solve for 'an expression in the integral that contains dx' instead, like this:
[tex]\begin{aligned}
u &= x^8 \\
du &= 8x^7 dx \\
\frac{1}{8} du &= x^7 dx
\end{aligned}[/tex]
(because x7 dx appears in the integral)

So,
[tex]\int x^{7}e^{x^{8}}dx = \frac{1}{8}\int e^u du = ...[/tex]
 
  • #11
Thank you both so much!
And to eumyang, I don't understand, I thought it would be easier to isolate the dx, because you can simply just substitute the dx, isn't that right?

Anyway, thanks again!
 

1. What is an indefinite integral?

An indefinite integral is a mathematical operation that finds the most general antiderivative of a given function. It is represented by the symbol ∫, and is the reverse operation of differentiation.

2. Why do I get stuck when evaluating an indefinite integral?

Evaluating indefinite integrals can be complex and require a thorough understanding of calculus principles. Sometimes, the integrand may not have a closed-form solution, making it difficult to find the antiderivative. Additionally, a mistake in the integration process can also lead to being stuck.

3. How can I approach evaluating an indefinite integral?

There are several techniques that can be used to evaluate indefinite integrals, such as substitution, integration by parts, and partial fractions. It is important to identify which technique would be most appropriate for the given integrand, and to carefully follow the steps to find the antiderivative.

4. Can indefinite integrals be solved using a calculator?

Yes, many scientific calculators have a built-in function to evaluate indefinite integrals. However, it is still important to have a basic understanding of the concepts and techniques involved in solving these integrals.

5. How can I check if my answer to an indefinite integral is correct?

You can check your answer by taking the derivative of the antiderivative you found. If the resulting function is the original integrand, then your answer is correct. Additionally, you can also use online integration tools or ask for help from a math tutor or teacher.

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