Some very tough integration problems I am stuck on.

In summary: I'm having some trouble with the integration of arctan, too. I think I need to integrate by parts and use the substitution u = sqrt(x).Here is how that would look:\int u^{n}du\begin{align*}& u^{n}& & & = & \frac{1}{2} \int arctan(u) du\\& & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & &
  • #1
nickclarson
32
0
[Solved] Some very tough integration problems I am stuck on.

Got these for homework the other day and they are due today! I am having trouble on where to start with all of them.

[tex]\int x^{n}lnx dx[/tex]

[tex]\int sin(\sqrt{x}) dx[/tex]

don't even know where to start on this one... should I substitute for sqrt(x)?

[tex]\int xarctan(x^{2}-1) dx[/tex]

I think I should substitute for what's inside arctan, but then I'm left with [tex]\frac{1}{2} \int arctan(u) dx[/tex] and I don't know how to integrate arctan
 
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  • #2
I'll give you a hint for the first one: Integration by parts, but be careful which one you chose to differentiate (which one simplifies on differentiation?).
 
  • #3
Ok that's what I thought for the first one. We were taught to pick the one that won't change for the part that we integrate... and the other for the part that we differentiate.

So I would pick lnx to differentiate? OHH! I just figured the first one out! Thanks so much!

Still stuck on the others though.
 
  • #4
nickclarson said:
OHH! I just figured the first one out!
Well done :approve:

For the second one, how about a substitution (followed by integration by parts)?
 
  • #5
Ok This is what I did for the second one:

I picked sqrt(x) as u.

[tex]2 \int usin(u) du[/tex]

The integration by parts is what I has me confused. I know this is the form:

[tex]uv - \int v du[/tex]

I'm not sure what to pick as my u and dv, my teacher didn't explain it very well in class. I think I should pick sin(u) for my dv though and the rest for my u (or in this case t since I already used u for my substitution). Am I on the right track?

Thanks for all of your help by the way. :)
 
  • #6
nickclarson said:
Ok This is what I did for the second one:

I picked sqrt(x) as u.

[tex]2 \int usin(u) du[/tex]

The integration by parts is what I has me confused. I know this is the form:

[tex]uv - \int v du[/tex]

I'm not sure what to pick as my u and dv, my teacher didn't explain it very well in class. I think I should pick sin(u) for my dv though and the rest for my u (or in this case t since I already used u for my substitution). Am I on the right track?

Thanks for all of your help by the way. :)

well, generally there is no one pattern on how to approach these problems, i mean on which one to chose as v and which as du, it comes with experience. Meanwhile some trial and error would be just fine.
 
  • #8
Wow, perfect thank you very much. Wish my teacher would have shown me that one.
 
  • #9
Here are my solutions for the first two:

[tex]1. \left[ \frac{xlnx}{n+1} - \frac{x}{(n+1)^{2}} \right] x^{n}[/tex]

[tex]2. 2sin(\sqrt{x}) - 2\sqrt{x}cos(\sqrt{x})[/tex]

I think that's right for #2

3. Still no go
 
  • #10
Here are my solutions for the first two:

1. [tex]\left[ \frac{xlnx}{n+1} - \frac{x}{(n+1)^{2}} \right] x^{n}[/tex]

2. [tex]2\left[sin(\sqrt{x}) - \sqrt{x}cos(\sqrt{x})\right][/tex]

I think that's right for #2

3. Still no go
 
  • #11
alrighty brother, i will show you the techniques i learned to solve these puppies.

Integration by parts w/ borrowing:
1. Separate the problem into two parts a derivative(x) and an antiderivative(y/dx)
2. The Derivative must approach zero
3. Because the antiderivative of Lnx is DUMB, we must always take the derivative(x) of it.

[tex]\int[/tex]x[tex]^{n}[/tex]lnxdx

u ___________ dv

lnx __________ x[tex]^{n}[/tex]

Continue to take the deriv/antideriv until you are back to where you began ( in this case lnx) and ignore sign changes(+-)

u ____________ dv

lnx___________ x[tex]^{n}[/tex]
1/x___________ x[tex]^{v+1}[/tex]/(n+1)
lnx___________ Whatever this is

add each of the parts diagonally varying signs(+/-)

u ___________dv

lnx <--------] x[tex]^{n}[/tex]
1/x <______ ] [+]------> x[tex]^{v+1}[/tex]/(n+1)
lnx [[-] ________> whatever this is

lnx+x[tex]^{v+1}[/tex]/(n+1)[/tex] - ( 1/x + whatever this is) write them out, distribute, add everything together, cancel out terms and simplify.
 
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  • #13
i also have a problem in finding the integration of "inetgration of square root of sinx "...and even "x square divided by 1+ x power 5"
 

Related to Some very tough integration problems I am stuck on.

1. Why are integration problems important in science?

Integration problems are important in science because they allow us to model and understand complex phenomena in a quantitative manner. They also help us to make predictions and analyze data, which are essential for scientific research.

2. What makes some integration problems particularly difficult?

Some integration problems can be difficult due to their complexity, involving multiple variables and functions. They may also require advanced mathematical techniques and a deep understanding of the underlying principles.

3. How can I approach tough integration problems?

One approach is to break the problem down into smaller, more manageable parts. It can also be helpful to visualize the problem and try different techniques, such as substitution or integration by parts. Additionally, seeking help from a tutor or colleague can provide valuable insights.

4. Are there any common mistakes to watch out for in integration problems?

Yes, some common mistakes include not considering all the variables and limits, not using the correct integration techniques, and making algebraic errors. It is important to always double-check your work and go back to the problem statement to ensure all aspects have been addressed.

5. How can I improve my integration problem-solving skills?

Practice, practice, practice! The more you work through integration problems, the more familiar you will become with the techniques and strategies. It is also helpful to review the fundamental concepts and principles of integration, as well as seeking feedback and guidance from others.

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