Integration using u substitution and arctan

In summary, the conversation discusses a problem with finding the coefficients in a given equation. The main chunk of the problem is expressed as (a)ln|(z-4)^2+25|+(b)arctan((z-4)/5), with a and b equal to 5 and 32/5, respectively. The main issue is splitting up the top portion of the equation so that the first part is the derivative of the bottom and the second part is the correct constant. The individual is seeking further explanation on how to find these coefficients.
  • #1
dante3001
2
0
so I'm having problems with the coefficients in this problem.

[tex]\int[/tex](10z+8/z^2-8z+41)dz

i know that the main chunk is

(a)ln|(z-4)^2+25|+(b)arctan((z-4)/5)

a and b are 5 and 32/5 respectively
the problem is i can't seem to split up the top so that the first portion is the derivitive of the bottom and that the other top is the right constant.
 
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  • #2
(a)ln|(z-4)^2+25|+(b)arctan((z-4)/5)

If this is not the answer how it is related to the question..?:smile:
 
  • #3
the main part is not what i need explained which is the chunk there. it is the a and b portion. i just need to know how the coefficients are found
 
  • #4
dante3001 said:
[tex]\int[/tex](10z+8/z^2-8z+41)dz

Do you mean
[tex]\int\frac{10z+8}{z^2-8z+41}dz[/tex]
?

To me it is not clear what you already solved and where your problem is. I suggest you present what you did so far and how you obtained the partial answer (a).

-Pere
 
Last edited:

1. What is u substitution in integration and why is it useful?

U substitution is a technique used in integration to simplify integrals by replacing a variable with a new variable, u. This allows for the use of simpler integration rules and often leads to a more manageable integral. It is useful because it can be used to solve a wide range of integrals that would otherwise be difficult or impossible to solve.

2. How do I know when to use u substitution in integration?

There are several signs that may indicate the use of u substitution in integration. These include integrals involving products or quotients of functions, expressions with powers of trigonometric functions, and integrals with square root terms. If you are unsure, try substituting u and see if it simplifies the integral.

3. What is the process for using u substitution in integration?

The process for using u substitution in integration involves the following steps:

  1. Select an appropriate substitution, u = g(x).
  2. Calculate du/dx.
  3. Substitute u and du/dx into the integral, replacing all occurrences of x and dx.
  4. Simplify the integral using basic integration rules.
  5. Replace u with the original variable, x.

4. What is the role of arctan in integration using u substitution?

Arctan, or inverse tangent, is often used in integration using u substitution when dealing with integrals involving inverse trigonometric functions. This is because the derivative of arctan is 1/(1+x^2), which can be useful in simplifying the integral. By substituting u = arctan(x), we can often rewrite the integral in a simpler form.

5. Are there any common mistakes to watch out for when using u substitution in integration?

Yes, there are a few common mistakes to watch out for when using u substitution in integration. These include forgetting to include the derivative, du/dx, when substituting, using the wrong substitution, and forgetting to replace u with the original variable after simplifying the integral. It is important to carefully follow the steps and double check your work to avoid these mistakes.

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