Integral of 1/u^n knowing integral of 1/u

[ezintegral]

Hello,

This is the initial problem:

Find the primitive of 1/(x4+1)

i've done this, the value of the primitive is this ugly looking expression:

http://image.bayimg.com/d9ac5052a83d5808955a7a647c6cb7343f9ace1f.jpg

Now the question asked is to deduce the primitive value of 1/(x4+1)^3 from what I found.

This is why I'm asking if there is any general method to compute the value of ∫1/u^n knowing ∫1/u

Thank you

 

[dextercioby]

No, essentially there isn't. $\int \frac{1}{(1+x^4)^3}$ is solved through partial fractions.

 

[Mark44]

Hello,

This is the initial problem:

Find the primitive of 1/(x4+1)

i've done this, the value of the primitive is this ugly looking expression:

http://image.bayimg.com/d9ac5052a83d5808955a7a647c6cb7343f9ace1f.jpg

Your image isn't showing.

Now the question asked is to deduce the primitive value of 1/(x4+1)^3 from what I found.

This is why I'm asking if there is any general method to compute the value of ∫1/u^n knowing ∫1/u

Two comments here:
1. The two formulas are not related.
$$ \int \frac{du}{u^n} = \int u^{-n}du = \frac{u^{-n + 1}}{-n + 1} + C$$
$$\int \frac{du}{u} = ln|u| + C$$

The integrals you show don't fit either of these formulas, because if u = x⁴ + 1, then du = 4x³dx, which you don't have.

2. You have omitted the differential du from your integrals above. This isn't so crucial in the early stages of learning integration, but it is for more complicated methods such as integration by parts and trig substitution.

 

[ezintegral]

Thanks for the answers.

I'm going to try rewriting 1/(x4+1) in partial equations and see what I can get.

 

[Mark44]

Thanks for the answers.

I'm going to try rewriting 1/(x4+1) in partial equations and see what I can get.

Presumably you mean 1/(x⁴ + 1). At the very least, use '^' to write this as 1/(x^4 + 1).

As for breaking up 1/(x⁴ + 1) using partial fractions, I don't see how this will do you any good. The denominator cannot be reduced to lower-degree factors with real coefficients.

It would help if you showed us the exact problem you're working on.

 

[SammyS]

... The denominator cannot be reduced to lower-degree factors with real coefficients.
...

x⁴ + 1 can be reduced to lower-degree factors with real coefficients, but not to lower-degree factors with integer coefficients.

$x^4+1=(x^2+(\sqrt{2})x+1)(x^2-(\sqrt{2})x+1)$

 

[ezintegral]

Presumably you mean 1/(x⁴ + 1). At the very least, use '^' to write this as 1/(x^4 + 1).

As for breaking up 1/(x⁴ + 1) using partial fractions, I don't see how this will do you any good. The denominator cannot be reduced to lower-degree factors with real coefficients.

It would help if you showed us the exact problem you're working on.

This problem is presented in 2 questions.

1) Find the primitive of $\frac{1}{x^4 + 1}$

This is done, no particular problem. You get this:

http://image.bayimg.com/d9ac5052a83d5808955a7a647c6cb7343f9ace1f.jpg


2) Deduce the primitive of $\frac{1}{(x^4 + 1)^3}$ from the primitive of $\frac{1}{x^4 + 1}$


Now the problem is with 2). I don't see how I can use the first result to find the second one.

Thank you for your time.

 

[Mark44]

x⁴ + 1 can be reduced to lower-degree factors with real coefficients, but not to lower-degree factors with integer coefficients.

$x^4+1=(x^2+(\sqrt{2})x+1)(x^2-(\sqrt{2})x+1)$

Thanks...

 

[Mark44]

This problem is presented in 2 questions.

1) Find the primitive of $\frac{1}{x^4 + 1}$

This is done, no particular problem. You get this:

http://image.bayimg.com/d9ac5052a83d5808955a7a647c6cb7343f9ace1f.jpg


2) Deduce the primitive of $\frac{1}{(x^4 + 1)^3}$ from the primitive of $\frac{1}{x^4 + 1}$


Now the problem is with 2). I don't see how I can use the first result to find the second one.

Thank you for your time.

The image above is attached here.

 

[dextercioby]

$$I_{n} = \int\frac{dx}{\left(1+x^4\right)^n}, \\ ~, n\in \mathbb{N}$$

Point 1 asks you to compute I₁. I₃ follows from twice part integration.

I_n is expressed through a Gauss hypergeometric function as

http://www.wolframalpha.com/input/?i=Evaluate+integral+of+1%2F%281%2Bx^4%29^n+

 

[gabbagabbahey]

$$I_{n} = \int\frac{dx}{\left(1+x^4\right)^n}, \\ ~, n\in \mathbb{N}$$

Point 1 asks you to compute I₁. I₃ follows from twice part integration.

Integration by parts once is enough to determine a recursive relationship of the form $I_{n+1}(x)=f(x, n)I_n(x)+g(x, n)$. $I_3$ can then be expressed in terms of $I_1(x)$ by using that relationship twice.

 

[ezintegral]

Integration by parts once is enough to determine a recursive relationship of the form $I_{n+1}(x)=f(x, n)I_n(x)+g(x, n)$. $I_3$ can then be expressed in terms of $I_1(x)$ by using that relationship twice.

Thank you, this looks promising and within my reach.