Troubleshooting Integral: Partial Fractions with Unsolvable Component"

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In summary, the integral of (x^2 + 1) / (x^3 + 8) dx can be solved using partial fractions and a substitution, with the final answer involving ln and tan^-1 functions. The substitution process may require rearranging expressions using trigonometric identities.
  • #1
sony
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Ok, so I have the integral of (x^2 + 1) / (x^3 + 8) dx

I use partial fractions and end up with two integrals, the one I cannot solve is:

1/12 * "integral of" (7x-4) / [(x-1)^2 +3 ] dx
With u=x-1 I get
(7u-3) / (u^2+3) du

But I have no Idea of how to solve it, the answer in the books shows the step up to this, but jumps over what to do with the last part and writes the answer (something involving tan^-1 and ln)

Please help!
Thanks
 
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  • #2
sony said:
Ok, so I have the integral of (x^2 + 1) / (x^3 + 8) dx

I use partial fractions and end up with two integrals, the one I cannot solve is:

1/12 * "integral of" (7x-4) / [(x-1)^2 +3 ] dx
With u=x-1 I get
(7u-3) / (u^2+3) du

But I have no Idea of how to solve it, the answer in the books shows the step up to this, but jumps over what to do with the last part and writes the answer (something involving tan^-1 and ln)

Please help!
Thanks
Factor out the 3 in the denominator:

[tex]\int\frac{7u-3}{u^2+3}\,du=\frac{1}{3}\int\frac{7u-3}{\left(\frac{u}{\sqrt{3}}\right)^2+1}\,du[/tex]

Then make the subsititution [itex]u=\sqrt{3}\tan{\theta}[/itex].

Alex
 
  • #3
Oh, thanks!
 
  • #4
Ok I'm very confused about the doing substitution _two_ times, can someone please go threw every step of this?
 
  • #5
Fra børjan av?
 
  • #6
Sorry, nevermind! I think I can figure this out, can someone please just tell me what the integral of Tan[x] equals?
 
  • #7
It is C-ln(|cos(x)|), where C is an integration constant.
 
  • #8
Ok, I see it now. Took some time :P

Thanks!
 
  • #9
Ok, bah. I don't get it. Can someone please take it from what apmcavoy wrote (start with subsitution of u)
 
  • #10
You have then:
[tex]du=\frac{\sqrt{3}d\theta}{\cos^{2}\theta}[/tex]
Since [tex]tan^{2}\theta+1=\frac{1}{\cos^{2}\theta}[/tex]
we get:
[tex]\frac{1}{3}\int\frac{7u-3}{(\frac{u}{\sqrt{3}})^{2}+1}du=\frac{1}{3}\int\sqrt{3}(7tan\theta-3)d\theta[/tex]
All right?
 
  • #11
Yeah, thanks. I made it to the last step, but I don't see what happens with the Cos bit... :P

(EDIT: The cos bit in the solution of the integral of Tan phi)
 
  • #12
Okay, you'll basically need to rearrange an expression like cos(Atan(y)), which will appear within the logarithm.

In order to do this, not that by definition of the Atan and tan functions, we have:
[tex]y=\tan{Atan(y)}=\frac{\sin(Atan(y))}{\cos(Atan(y))}=\frac{\sqrt{1-\cos^{2}(Atan(y))}}{\cos(Atan(y))}(1)[/tex]
where I've used the identity [itex]\sin^{2}x=1-\cos^{2}x[/itex] for all x.

Thus, from (1), we get [tex]y^{2}\cos^{2}(Atan(y))=1-\cos^{2}(Atan(y))[/tex]
by which we have:
[tex]|\cos(Atan(y))|=\frac{1}{\sqrt{1+y^{2}}}[/tex]

Thus, we have rewritten the troublesome expression, in that we now have:
[tex]ln|\cos(Atan(y))|=-\frac{1}{2}ln(y^{2}+1)[/tex]
 
  • #13
Ok, this made it clear. Thank you!
 
  • #14
You're welcome.
 

FAQ: Troubleshooting Integral: Partial Fractions with Unsolvable Component"

What is an integral?

An integral is a mathematical concept that represents the area under a curve in a graph. It is often used in calculus to find the total value of a function over a given interval.

What are partial fractions?

Partial fractions are a method of breaking down a complex rational function into simpler fractions. It is useful in integrating functions that cannot be easily solved using traditional methods.

What is an unsolvable component in partial fractions?

An unsolvable component in partial fractions refers to a fraction that cannot be further simplified. This often occurs when the denominator of the fraction cannot be factored into linear factors.

Why is troubleshooting integral: partial fractions with unsolvable component important?

Troubleshooting integral: partial fractions with unsolvable component is important because it allows for the integration of functions that cannot be solved using traditional methods. It also helps in finding the most simplified form of a function, which can make further calculations and analyses easier.

What are some tips for troubleshooting integral: partial fractions with unsolvable component?

Some tips for troubleshooting integral: partial fractions with unsolvable component include: thoroughly understanding the concept of partial fractions, identifying the unsolvable component, using algebraic manipulation to simplify the fraction, and checking your answer by differentiating the integrated function to ensure it matches the original function.

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