Primitive Function: Finding the Right Direction

In summary, the conversation discusses a problem with finding a primitive function and various attempts at solving it using substitutions and partial fractions decomposition. Ultimately, the method of setting t=tan(x) and using partial fractions is found to be successful.
  • #1
twoflower
368
0
Hi,
I don't know how to find this primitive function:

[tex]
\int \frac{dx}{(1+\tan x)(1+\tan^2 x)}
[/tex]

I tried substitutions [itex]t = \tan x[/itex] or [itex]t = 1 + \tan x[/itex], but it didn't seem to help me lot...

Could someone please point me to the right direction?

Thank you.
 
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  • #2
1. Set [tex]t=tan(x)\to\frac{dt}{dx}=\frac{1}{\cos^{2}(x)}=tan^{2}x+1\to{dx}=\frac{dt}{t^{2}+1}[/tex]
Thus, you've got:
[tex]\int\frac{dx}{(1+tan(x))(1+tan^{2}x)}=\int\frac{dt}{(1+t)(t^{2}+1)^{2}}[/tex]
This can be solved by partial fractions decomposition.
 
  • #3
Thank you arildno, I made a mistake that I didn't simply change tan x = t and dx = dt/t^2 + 1, instead I expressed tan x as sin x / cos x and divided the denominator with cos^2 x and it turned into crazy powers of t. Your method works great, thanks.
 

1. What is a primitive function?

A primitive function is a basic mathematical function that cannot be simplified any further. It is the most fundamental building block of more complex mathematical functions and is typically represented as a single variable.

2. How do you find the primitive function?

To find the primitive function of a given function, you must use integration, which is the inverse operation of differentiation. This involves finding an expression that, when differentiated, gives the original function.

3. Why is finding the primitive function important?

Finding the primitive function is important in many fields such as physics, engineering, and economics. It allows us to calculate important quantities such as distance, velocity, and acceleration, as well as areas and volumes under curves.

4. What are some common techniques for finding the primitive function?

Some common techniques for finding the primitive function include using tables of known integrals, using integration by substitution, and using integration by parts. These techniques require a good understanding of algebra, trigonometry, and calculus.

5. Are there any tools or software that can help with finding the primitive function?

Yes, there are several tools and software that can help with finding the primitive function. Some popular options include WolframAlpha, Symbolab, and Desmos. These tools use algorithms and pre-programmed rules to find the primitive function of a given function.

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