Integrate Trig Expression: $\int_\frac{dx}{(4+x^2)^2}dx$

In summary, the given integral can be simplified to cos²(x) and can be further simplified by using substitution and integration by parts.
  • #1
Aerosion
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Homework Statement



[tex]\int_\frac{dx}{(4+x^2)^2}dx[/tex]

Homework Equations





The Attempt at a Solution



SO...I started by making [tex]x = 2*tan x[/tex] and [tex]dx = 2*sec(x)^2[/tex]. The x is supposed to be the 0 sign with the line through it, but I don't know how to make that.

I then made the equation [tex]\int \frac{2*sec(x)^2}{(4+2*tan(x)^2}*2*sec(x)^2[/tex]. I multiplied the two secants to get [tex]4*sec(x)^4[/tex] on the top, and then I turned the [tex]2*tan(x)^2[/tex] on the bottom into [tex]2*sec(x)^2-2[/tex]. The equation now looks like [tex]\int \frac{4*sec(x)^4}{(4+2*sec(x)^2-2}[/tex]. How does this simplify? I want to get rid of the [/tex]2*sec(x)^2[/tex] by dividing it with the top thing, but I don't think I can do that becaause of the -2 attached to it.
 
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  • #2
you should fix your latex formatting.

your integral becomes cos²(x).

Show your work again, and let us know if you get stuck.
 
  • #3
[tex] \int \left(\frac{1}{-2x}\right)\left(\frac{-2x}{(4+x^2)^2}{}dx\right)=-\frac{1}{2x}\cdot\frac{1}{4+x^2}+\frac{1}{2}\int \frac{dx}{x^2 (4+x^2)} [/tex]

[tex] =-\frac{1}{2x}\cdot\frac{1}{4+x^2}+\frac{1}{8}\int \left(\frac{1}{x^2}-\frac{1}{4+x^2}\right){}dx [/tex]

...
 

1. What is the method for integrating trigonometric expressions?

The method for integrating trigonometric expressions involves using trigonometric identities and substitution techniques to simplify the expression and solve for the integral.

2. How do you integrate expressions with a trigonometric function in the denominator?

To integrate expressions with a trigonometric function in the denominator, use the substitution method by letting u = the trigonometric function in the denominator and then solving for the integral in terms of u.

3. Can you provide an example of integrating a trigonometric expression?

One example of integrating a trigonometric expression is the integral of $\int_\frac{sin(x)}{cos(x)}dx$, which can be solved by using the substitution method with u = cos(x) and solving for the integral in terms of u.

4. Is it possible to integrate an expression with multiple trigonometric functions?

Yes, it is possible to integrate an expression with multiple trigonometric functions by using the trigonometric identities and substitution techniques to simplify the expression and solve for the integral.

5. What is the general rule for integrating trigonometric expressions?

The general rule for integrating trigonometric expressions is to use substitution techniques and trigonometric identities to simplify the expression and solve for the integral in terms of the substituted variable.

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