Integral of $\frac{1}{(3+4\sin x)^2}dx$

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SUMMARY

The integral of $\frac{1}{(3+4\sin x)^2}dx$ can be solved using the substitution $t= \tan \frac{x}{2}$, leading to $dx= \frac{2}{1+t^{2}} dt$ and $\sin x=\frac{2 t}{1+t^{2}}$. This substitution simplifies the integral, although the resulting expression is not straightforward. Wolfram Alpha can provide both the steps and the final answer for this integral, confirming the validity of the substitution method proposed.

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$\displaystyle \int\frac{1}{(3+4\sin x)^2}dx$
 
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Re: defeinite Integral

jacks said:
$\displaystyle \int\frac{1}{(3+4\sin x)^2}dx$

The 'standard' substition for this type of integral is...

$\displaystyle t= \tan \frac{x}{2} \implies x=2\ \tan^{-1} t \implies dx= \frac{2}{1+t^{2}}\ dt \implies \sin x=\frac{2 t}{1+t^{2}} \implies \cos x= \frac{1-t^{2}}{1+t^{2}}$

Kind regards

$\chi$ $\sigma$
 
Re: defeinite Integral

jacks said:
$\displaystyle \int\frac{1}{(3+4\sin x)^2}dx$

definite/indefinite integral?

This does have an (indefinite) integral in terms of elementary functions, but it is not particularly simple (at least if you assume Wolfram Alpha has chosen a good approach to doing this, it starts with the substitution chisigma proposes in his post. Alpha will give you the steps as well as the final answer so you may as well ask the horses mouth itself)

CB
 

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