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

In summary, the conversation discusses the definite integral, $\displaystyle \int\frac{1}{(3+4\sin x)^2}dx$, and a possible substitution method, $\displaystyle t=\tan\frac{x}{2}$, to solve it. The response also mentions that the integral can be solved using elementary functions, but it may not be a simple solution.
  • #1
juantheron
247
1
$\displaystyle \int\frac{1}{(3+4\sin x)^2}dx$
 
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  • #2
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

$\chi$ $\sigma$
 
  • #3
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
 

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

What is the integral of $\frac{1}{(3+4\sin x)^2}dx$?

The integral of $\frac{1}{(3+4\sin x)^2}dx$ is a function that represents the area under the curve $\frac{1}{(3+4\sin x)^2}$ with respect to the variable $x$. It can be evaluated using various techniques, such as substitution or integration by parts.

Why is the integral of $\frac{1}{(3+4\sin x)^2}dx$ important?

The integral of $\frac{1}{(3+4\sin x)^2}dx$ is important in various fields of science, including physics, engineering, and mathematics. It is used to solve problems involving periodic functions and can provide valuable insights into the behavior of systems that exhibit periodicity.

Can the integral of $\frac{1}{(3+4\sin x)^2}dx$ be evaluated analytically?

Yes, the integral of $\frac{1}{(3+4\sin x)^2}dx$ can be evaluated analytically using various techniques. However, in some cases, it may not have an exact closed-form solution and may need to be approximated using numerical methods.

How can the integral of $\frac{1}{(3+4\sin x)^2}dx$ be applied in real-world problems?

The integral of $\frac{1}{(3+4\sin x)^2}dx$ has many applications in real-world problems, such as calculating the displacement, velocity, or acceleration of an object moving in a circular motion. It is also used in signal processing, digital image processing, and control systems.

Are there any special cases or restrictions when evaluating the integral of $\frac{1}{(3+4\sin x)^2}dx$?

Yes, there are some special cases and restrictions when evaluating the integral of $\frac{1}{(3+4\sin x)^2}dx$. For example, if the function is undefined at certain values of $x$ (such as when the denominator is equal to 0), the integral may need to be evaluated using different methods. Additionally, the interval of integration may need to be adjusted to avoid singularities or discontinuities in the function.

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