(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

By using the substitution t = tan x, find

[tex]\int \frac{dx}{\cos^2 x+4\sin^2 x}[/tex]

2. Relevant equations

3. The attempt at a solution

Well let tan x=t

[tex]\frac{dt}{dx}=\sec^2 x=\tan^2 x+1=1+t^2[/tex]

the integral then becomes

[tex]\int \frac{dx}{\frac{1}{\sqrt{1+t^2}}^2+4\frac{t}{\sqrt{1+t^2}}^2}[/tex]

which simplifies to

[tex]\int \frac{1}{1+4t^2}[/tex]

Then from here i make another substitution **

let t= 1/2 tan b

dt/db = 1/2 sec^2 b

[tex]\int \frac{1}{1+4(\frac{1}{2} \tan b)^2} \cdot \frac{1}{2}\sec^2 b db[/tex]

= b + constant

Back substitute

= [tex]\frac{1}{2}\tan^{-1} (2t)[/tex] + constant

= [tex]\frac{1}{2}\tan^{-1}(2\tan x)[/tex] + constant

Am i correct? Especially this part ** where i made another substitution, is that valid? Or when the question specified the substitution, i have to stick that one substitution only?

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# Homework Help: Can I use only one substitution for integral?

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