Solving Integrals with Fractions to U-Substitution Method

[MickGrif]

Hi, I'm trying to understand how you integrate fractions, I have a couple of examples which I was given, unfortunately the lecturer skipped the actual method and just gave us the answers.
By any chance could someone go through the method to solve these? (I think you use U substitution?)
Thanks in advance

(a) integral ((1-z)/(z2 +1))dz

(b) integral ((1)/(z+z2)) dz

(c) integral ((z)/(1-3z2))dz

I tried taking taking the fractions up and solving by parts but that just seems to complicate things further

 

[Curious3141]

Hi, I'm trying to understand how you integrate fractions, I have a couple of examples which I was given, unfortunately the lecturer skipped the actual method and just gave us the answers.
By any chance could someone go through the method to solve these? (I think you use U substitution?)
Thanks in advance

(a) integral ((1-z)/(z2 +1))dz

(b) integral ((1)/(z+z2)) dz

(c) integral ((z)/(1-3z2))dz

I tried taking taking the fractions up and solving by parts but that just seems to complicate things further

You're supposed to show more work, but I can give you some hints to get you started.

(a) Split up the numerator. The first fraction \frac{1}{z^2 + 1} can be immediately integrated if you recognise it as a special form. Alternatively, substitute z = \tan \theta. The second fraction -\frac{z}{z^2 + 1} can be integrated by substituting y = z^2 + 1.

(b) Factorise the denominator. Do a partial fraction decomposition.

(c) Substitute y = 1 - 3z^2.

 

[SammyS]

Hi, I'm trying to understand how you integrate fractions, I have a couple of examples which I was given, unfortunately the lecturer skipped the actual method and just gave us the answers.
By any chance could someone go through the method to solve these? (I think you use U substitution?)
Thanks in advance

(a) integral ((1-z)/(z2 +1))dz

(b) integral ((1)/(z+z2)) dz

(c) integral ((z)/(1-3z2))dz

I tried taking taking the fractions up and solving by parts but that just seems to complicate things further

Hello MickGrif. Welcome to PF !

As Curious3141 points out, the rules of this Forum require you to show your work before we can help you.

 

[klondike]

Is z real or complex? Which class are you getting these integrals?

z usually means α+jω (where jj=-1) in Complex Analysis. The integrals you listed would be typical elementary examples of application of Cauchy's residue theorem meant to show how to recognize the order of poles.

On the other hand, if z is real (weird convention to me), you have been given good hints.

Hi, I'm trying to understand how you integrate fractions, I have a couple of examples which I was given, unfortunately the lecturer skipped the actual method and just gave us the answers.
By any chance could someone go through the method to solve these? (I think you use U substitution?)
Thanks in advance

(a) integral ((1-z)/(z2 +1))dz

(b) integral ((1)/(z+z2)) dz

(c) integral ((z)/(1-3z2))dz

I tried taking taking the fractions up and solving by parts but that just seems to complicate things further

 

[Ray Vickson]

Hi, I'm trying to understand how you integrate fractions, I have a couple of examples which I was given, unfortunately the lecturer skipped the actual method and just gave us the answers.
By any chance could someone go through the method to solve these? (I think you use U substitution?)
Thanks in advance

(a) integral ((1-z)/(z2 +1))dz

(b) integral ((1)/(z+z2)) dz

(c) integral ((z)/(1-3z2))dz

I tried taking taking the fractions up and solving by parts but that just seems to complicate things further

If z2 is supposed to be z², you should either write it as z^2 or use the "X²" button on the palette at the top of this panel; that will give you z².

RGV

 

[MickGrif]

You're supposed to show more work, but I can give you some hints to get you started.

As Curious3141 points out, the rules of this Forum require you to show your work before we can help you.

Can I first thank you for the responses.
In regards showing the work done before this I have posted pics, the reason that I didnt do this in the first place was because the problem I was working on does not really tie into integrating up to this point. (As you will see below)

Is z real or complex? Which class are you getting these integrals?
z usually means α+jω (where jj=-1) in Complex Analysis. The integrals you listed would be typical elementary examples of application of Cauchy's residue theorem meant to show how to recognize the order of poles.
On the other hand, if z is real (weird convention to me), you have been given good hints.

The course is on ODEs (Ordinary Differential Equations) these problems in particular relating to 1st order homogenous ODEs

If z2 is supposed to be z², you should either write it as z^2 or use the "X²" button on the palette at the top of this panel; that will give you z².

RGV

I'm sorry it was meant to come out as z^2, I could have swore I clicked the button but must have double clicked or something, my bad.Work done up to this point; (alot of my own notes are in the margins so please excuse any thing that seems like obvious material :|)

(a)

(b)

(c)

 

[SammyS]

Hi, I'm trying to understand how you integrate fractions, I have a couple of examples which I was given, unfortunately the lecturer skipped the actual method and just gave us the answers.
By any chance could someone go through the method to solve these? (I think you use U substitution?)
Thanks in advance

(a) integral ((1-z)/(z2 +1))dz

(b) integral ((1)/(z+z2)) dz

(c) integral ((z)/(1-3z2))dz

I tried taking taking the fractions up and solving by parts but that just seems to complicate things further

For (a), split into two integrals.

\displaystyle \int\frac{1}{z^2+1}\,dz is a standard integral.

For \displaystyle \int\frac{z}{z^2+1}\,dz\,, use a substitution.​

For (b), use partial fraction decomposition.

\displaystyle \frac{1}{z+z^2}=\frac{A}{1+z}+\frac{B}{z}​

(c) can be done with a substitution.

 

[MickGrif]

For (a), split into two integrals.

\displaystyle \int\frac{1}{z^2+1}\,dz is a standard integral.

For \displaystyle \int\frac{z}{z^2+1}\,dz\,, use a substitution.​

For (b), use partial fraction decomposition.

\displaystyle \frac{1}{z+z^2}=\frac{A}{1+z}+\frac{B}{z}​

(c) can be done with a substitution.

For (a) could you use partial fractions as well? Or does the constant have to be positive, for example you can do ∫\frac{x+1}{(x+3)(x+2} with partial fractions, can you also do ∫\frac{x-1}{(x+3)(x+2)} using this method and thus use the same method for (a)?

 

[HallsofIvy]

If you are doing these as part of a Differential Equations class, you should be able to do these integrals easily. The questions you are asking are basic integral Calculus. For one thing, do you know what the derivative of arctan(x) is?

 

[MickGrif]

If you are doing these as part of a Differential Equations class, you should be able to do these integrals easily. The questions you are asking are basic integral Calculus. For one thing, do you know what the derivative of arctan(x) is?

That is extremely condescending. Clearly I can't do these easily, hence why I came looking for help. I didnt choose to do this module, it was not an option in my degree. All I want to do is to do ok in it.
Also I believe the derivative of arctan(x) is\frac{1}{1+x^{2}}

 

[SammyS]

...

Also I believe the derivative of arctan(x) is\frac{1}{1+x^{2}}

Yes, that's the derivative of the arctangent function .

 

[SammyS]

For (a) could you use partial fractions as well? Or does the constant have to be positive, for example you can do ∫\frac{x+1}{(x+3)(x+2} with partial fractions, can you also do ∫\frac{x-1}{(x+3)(x+2)} using this method and thus use the same method for (a)?

The problem with using partial fractions for (a), is that z²+1 is only factorable over the complex numbers:

z^2+1=(z-i)(z+i)\,.​

That would leave you with the logarithm of complex quantities.

 

[Millennial]

Take the principal branch of the logarithm and the definition of the complex logarithm yields the integral correctly (if it is definite.) Partial fractions decomposition is still applicable. The problem with it is that you would have to do some working with Euler's identity and the complex logarithm to get the expression in terms of real functions. With definite integrals, since you get a scalar, there is no such problem.

 

[HallsofIvy]

That is extremely condescending. Clearly I can't do these easily, hence why I came looking for help. I didnt choose to do this module, it was not an option in my degree. All I want to do is to do ok in it.
Also I believe the derivative of arctan(x) is\frac{1}{1+x^{2}}

Okay, what does that tell you about the integral of 1/(1+ x^2)?