integrate (x+2)/((x^2+2x+4)^2)
the usual partial fractions don't work???
(x+2)
--------------
(x+2)^4
=
1
--------------
(x+2)^3
sorry, factored wrong
I am at a loss. The denominator is repeated factors so normally one would use all powers up to the top one. However the denominator roots are complex and I get a real mess that is no easier than what we started with.
In some cases, the usual partial fraction decomposition method may not be applicable or may not simplify the integration process significantly. However, there are alternative techniques that you can use to solve this integral.
One possible approach is to use a trigonometric substitution. Let's consider the denominator: x^2 + 2x + 4. We can complete the square to rewrite it as (x + 1)^2 + 3. Now, we can make a substitution to simplify the integral further.
Let's substitute x + 1 = √3 tan(θ). By doing that, we can express both x and dx in terms of θ:
x = √3 tan(θ) - 1
dx = √3 sec^2(θ) dθ
Next, let's substitute x and dx in the integral:
∫(x + 2)/((x^2 + 2x + 4)^2) dx
= ∫((√3 tan(θ) - 1) + 2)/((√3 tan(θ) - 1 + 2)^2) √3 sec^2(θ) dθ
= ∫(√3 tan(θ) + 1)/(3 tan^2(θ))^2 √3 sec^2(θ) dθ
= ∫(√3 tan(θ) + 1)/(9 tan^4(θ)) √3 sec^2(θ) dθ
= ∫(√3 sec(θ) + sec^3(θ))/(9 tan^4(θ)) dθ
Now, you can integrate individual terms using standard integration techniques. For example, you can rewrite sec^3(θ) as sec(θ) * sec^2(θ) and use integration by substitution to solve it.
After integrating the individual terms, you will obtain the final result in terms of θ. Don't forget to convert back to the original variable, x, if necessary.
While this approach involves additional steps and calculations, it can be a viable alternative when the usual partial fraction decomposition method doesn't work well for a given integral.