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Primitive function

  • Thread starter twoflower
  • Start date
  • #1
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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.
 

Answers and Replies

  • #2
arildno
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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
368
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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.
 

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