Integrating a Tricky Devil: $\int \frac{\cos^5(\theta)}{\sin^4(\theta)} d\theta$

In summary, the conversation discusses a method for solving the integral \int \frac{\cos^5(\theta)}{\sin^4(\theta)} d\theta, using trigonometric identities to simplify the expression. The method involves rewriting the numerator as (1-sin^2(\theta))^2d(sin(\theta)). This simplification allows for integration by parts to solve the integral.
  • #1
gazzo
175
0
[tex]
\int \frac{\cos^5(\theta)}{\sin^4(\theta)} d\theta
[/tex]

Anyone mind sparing a little hint for this tricky devil? I can't even get started on it. [itex]\cot^4(\theta)\cos(\theta)[/itex] dosn't seem any better either.

I've tried using identities but I end up with nastier ones?
 
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  • #2
this is an easy one :

[tex]cos^5(\theta)d \theta = cos^4(\theta)cos(\theta) d \theta = (1-sin^2(\theta))^2d(sin(\theta))[/tex]

Fill this into the fraction and just work out the square in the numerator and integrate each part of the sum seperatly. If you want you can replace each sine by a dummy variable u but that is not necessary

marlon
 
  • #3
oooh yes of course!

Thank you very much.
 

Related to Integrating a Tricky Devil: $\int \frac{\cos^5(\theta)}{\sin^4(\theta)} d\theta$

1. What is the general approach for integrating a tricky devil?

The general approach for integrating a tricky devil is to first identify any possible trigonometric identities that can be used to simplify the expression. Next, try to manipulate the integrand to make it easier to integrate, such as using substitution or integration by parts. Finally, use the fundamental theorem of calculus to evaluate the integral.

2. How can I simplify the integrand in this expression?

In this specific case, you can use the power-reducing formula for cosine to rewrite $\cos^5(\theta)$ as $(\cos^2(\theta))^2 \cos(\theta)$. You can also use the reciprocal identity for sine to rewrite $\frac{1}{\sin^4(\theta)}$ as $(\csc^2(\theta))^2$. These simplifications will make the integral easier to evaluate.

3. Is there a specific substitution that can be used for this integral?

Yes, you can use the substitution $u = \sin(\theta)$ to simplify the integrand. This will result in the integral becoming $\int \frac{\cos^5(\theta)}{\sin^4(\theta)} d\theta = \int \frac{-\cos^2(u)}{u^4} du$. From here, you can use integration by parts to further simplify the expression.

4. What is the final answer for this integral?

The final answer for this integral is $\frac{-1}{3\sin^3(\theta)} + C$, where $C$ is the constant of integration. However, this answer can also be expressed in terms of the original variable $x$ by substituting back in $\theta = \sin^{-1}(x)$.

5. Can this integral be solved using a calculator or computer program?

Yes, most calculators and computer programs have built-in functions for evaluating integrals, including tricky ones like this. However, it is always important to understand the steps and techniques used to solve the integral rather than relying solely on technology.

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