• Support PF! Buy your school textbooks, materials and every day products Here!

Three integrals

  • Thread starter dirk_mec1
  • Start date
  • #1
761
13

Homework Statement




[tex]
1)\ \int \frac{1}{(x^2+4)^3}\ \mbox{d}x
[/tex]

[tex]
2)\ \int \frac{x}{x^4+x^2+1}\ \mbox{d}x
[/tex]

[tex]
3)\ \int \frac{x e^x}{(x+1)^2}\ \mbox{d}x
[/tex]

The Attempt at a Solution



I got hints but I don't know how to use them.

1) use trigonometric substitution and double angle formulas.
Do I have to use [tex] u=cos( \theta) [/tex] and after substituting the double angle formulas what to do next?

2) complete square in numerator
But I'm missing a x^2 term!

3) substitute y-1 and split integrand
Split the term?
 

Answers and Replies

  • #2
378
2
1. I would say use x = tan(u)

plug in values for x and dx.. and simplify. Integrate with respect to u and then convert from u to x
 
  • #3
761
13
1. I would say use x = tan(u)

plug in values for x and dx.. and simplify. Integrate with respect to u and then convert from u to x
That gives a [tex] \frac{1}{ \sec^4(u) }[/tex] does this makes life easier?
 
  • #4
103
0
I think the 2nd one can be solved as follows:

[tex]{\int}xdx/(x^2+1) = 1/2\sdot{\int}dx^2/(x^2+1) [/tex] and the rest is a piece of cake or am I missing something?
 
  • #5
761
13
I think the 2nd one can be solved as follows:
or am I missing something?
Yes a x^4 -term.
 
  • #6
tiny-tim
Science Advisor
Homework Helper
25,832
249
1) use trigonometric substitution and double angle formulas.
Do I have to use [tex] u=cos( \theta) [/tex] and after substituting the double angle formulas what to do next?

2) complete square in numerator
But I'm missing a x^2 term!

3) substitute y-1 and split integrand
Split the term?
That gives a [tex] \frac{1}{ \sec^4(u) }[/tex] does this makes life easier?
Hi dirk_mec1! :smile:

1) Hint: [tex] \frac{1}{ \sec^4(u) }\,=\,cos^4u[/tex]

2) Hint: No … you're missing a units term.

3) Do the substitution anyway, and show us what you get. :smile:
 
  • #7
103
0
The last one:

[tex] 3)\ \int \frac{x e^x}{(x+1)^2}\ \mbox{d}x = {\int}xde^x/(x+1)^2 = {\int} (x+1)de^x/(x+1)^2 - de^x/(x+1)^2 [/tex]

The rest I leave for you. If I am not mistaken you should get [tex] e^x/(x+1) [/tex]

Sorry for my LaTeX skills. I haven't figured it out yet
 
  • #8
761
13
Hi dirk_mec1! :smile:
2) Hint: No … you're missing a units term.

3) Do the substitution anyway, and show us what you get. :smile:
Hi tiny tim,

1) Oh now I see how to do the first one!
2) what is a units term?
3) working on it right now.
 
  • #9
tiny-tim
Science Advisor
Homework Helper
25,832
249
2) what is a units term?
Sorry … I meant an ordinary number (with no x).

Subtract a number from the denominator, to make a perfect square. :smile:

(armis, nice LaTeX, but just give hints … unless the OP didn't follow the last hint! :smile: )
 
  • #10
103
0
yeah, sorry tiny-tim. Although often times it's kind of tricky to express my thoughts in english just verbaly but I'll try harder next time. Thanks god math notation is the same for everyone

By the way I know it's not the right place to ask but how does one use the hide feature?
 
  • #11
761
13
Sorry … I meant an ordinary number (with no x).

Subtract a number from the denominator, to make a perfect square. :smile:
You mean like: [tex] (x^2+1)^2 -x^2 [/tex]

Thanks god math notation is the same for everyone
Use \frac{}{} for fractions
 
  • #12
103
0
oh, I see. Thanks
 
  • #13
tiny-tim
Science Advisor
Homework Helper
25,832
249
You mean like: [tex] (x^2+1)^2 -x^2 [/tex]
No … like [tex] (x^2+1)^2 - 3 [/tex] (but different, of course!) :smile:
By the way I know it's not the right place to ask but how does one use the hide feature?
hmm … I was going to ask "what hide feature?" … and then I vaguely remembered seeing something about it … I think you type the full answer, but with a tag which hides it unless someone puts the mouse over it.

But I've no idea how to operate it.

You'd better ask someone more knowledgable than me! :redface:

Anyway, it's good practice only giving hints! :wink:
 
  • #14
761
13
No … like [tex] (x^2+1)^2 - 3 [/tex] (but different, of course!) :smile:
So you mean:

[tex] \left( x^2 + \frac{1}{2} \right)^2 +\frac{3}{4} [/tex]



So we get

[tex]
\int \frac{x}{x^4+x^2+1}\ \mbox{d}x = \int \frac{x}{ \left( x^2 + \frac{1}{2} \right)^2 +\frac{3}{4}}\ \mbox{d}x
[/tex]


Now...what to do next?
 
Last edited:
  • #15
tiny-tim
Science Advisor
Homework Helper
25,832
249
So you mean:

[tex] \left( x^2 + \frac{1}{2} \right)^2 +\frac{3}{4} [/tex]

So we get

[tex]
\int \frac{x}{x^4+x^2+1}\ \mbox{d}x = \int \frac{x}{ \left( x^2 + \frac{1}{2} \right)^2 +\frac{3}{4}}\ \mbox{d}x
[/tex]
:biggrin: Woohoo! :biggrin:
Now...what to do next?
Make the obvious substitution. :wink:
 
  • #16
761
13
:biggrin: Woohoo! :biggrin:

Make the obvious substitution. :wink:

Let [tex]u = x^2+\frac{1}{2} [/tex]


Then


[tex]
\int \frac{x}{x^4+x^2+1}\ \mbox{d}x = \int \frac{x}{ \left( x^2 + \frac{1}{2} \right)^2 +\frac{3}{4}}\ \mbox{d}x = \frac{1}{2} \cdot \int \frac{1}{u^2+\frac{3}{4}}\ \mbox{d}x = \frac{1}{2} \arctan \left( \frac{u}{\frac{1}{2} \sqrt{3}} \right)\ +\ C
[/tex]

Now put the u in terms of x back and we've got the answer. Is this okay Tim?
 
  • #17
tiny-tim
Science Advisor
Homework Helper
25,832
249
Yes, that's fine … except you've left out the (2/√3) outside the arctan.

(See Hootenanny's Library entry, List of Standard Integrals)

How did you get on with question 3? :smile:
 
  • #18
103
0
I am sorry to interrupt, but wasn't there a much easier way to solve the 2nd problem?

[tex]{\int}\frac{xdx}{(x^2+1)} = 1/2\sdot{\int}\frac{dx^2}{(x^2+1)}[/tex]
 
  • #19
761
13
Yes, that's fine … except you've left out the (2/√3) outside the arctan.
Whoops, you're right.

(See Hootenanny's Library entry, List of Standard Integrals)
I use the wiki-page :wink:

How did you get on with question 3? :smile:

[tex]
\int \frac{x e^x}{(x+1)^2}\ \mbox{d}x = \frac{x e^x}{(x+1)^2} - \int e^x \frac{(x+1)^2-2x^2-2x}{(x+1)^4}
[/tex]

This is getting nasty....

By the way I didn't understand Armis' hint :biggrin:
 
Last edited:
  • #20
761
13
I am sorry to interrupt, but wasn't there a much easier way to solve the 2nd problem?

[tex]{\int}\frac{xdx}{(x^2+1)} = 1/2\sdot{\int}\frac{dx^2}{(x^2+1)}[/tex]
Where did you get this integral? This isn't the second integral I posted, right?
 
  • #21
103
0
"By the way I didn't understand Armis' hint"

:) I'll explain it once you get the problem right
 
  • #22
103
0
Ah, sorrry

I meant [tex]{\int}\frac{xdx}{(x^2+1)^2} = 1/2\sdot{\int}\frac{dx^2}{(x^2+1)^2}[/tex]

Oh, but it's not the same in any case :( Sorry
 
  • #23
tiny-tim
Science Advisor
Homework Helper
25,832
249
This is getting nasty....
That's because you didnt follow the nice hint: :wink:
substitute y-1
(armis … it still doesn't match the original question, does it?)
 
  • #24
761
13
[tex]
\int \frac{x e^x}{(x+1)^2}\ \mbox{d}x = \frac{-xe^x}{x+1} + \int \frac{1}{x+1} \left( e^x+x e^x \right) \ \mbox{d}x =\frac{-xe^x}{x+1} + e^x = \frac{e^x}{x+1}[/tex]
 
  • #25
tiny-tim
Science Advisor
Homework Helper
25,832
249
[tex]
\int \frac{x e^x}{(x+1)^2}\ \mbox{d}x = \frac{-xe^x}{x+1} + \int \frac{1}{x+1} \left( e^x+x e^x \right) \ \mbox{d}x =\frac{-xe^x}{x+1} + e^x = \frac{e^x}{x+1}[/tex]
hmm … not the way they suggested … but it certainly works! :smile:

(but you'd better use their hints also, just for practice …
i] substitute y = x + 1
or ii] use partial fracitons :smile:)
 

Related Threads for: Three integrals

  • Last Post
Replies
4
Views
821
  • Last Post
Replies
3
Views
922
  • Last Post
Replies
10
Views
1K
Replies
0
Views
2K
Replies
4
Views
473
Replies
8
Views
1K
Top