What type of integration to use?

In summary, the conversation discusses the best approach to solving the integral ∫(x2+9x)/(4x2) dx. It starts with using trigonometric substitution, but ultimately simplifies the integral to (1/4)∫(x+9)/(x) dx. The experts suggest using algebraic manipulation and splitting up the numerator to solve the integral without the need for substitution.
  • #1
jdawg
367
2

Homework Statement


∫(x2+9x)/(4x2) dx


Homework Equations





The Attempt at a Solution


I started by using trigonometric substitution:
x2=9tan2θ
x=3tanθ
dx=3sec2θ dθ

∫9(sec2θ)/4(9tan2)*3sec2θ dθ

(3/4)∫(sec4θ)/(tan2θ) dθ

I'm not really sure what to do next, or if I even used the best method of integration. Could I maybe use u substitution and let tanθ=u? Or maybe rewrite the integral as (1/cos4θ)/(sin2θ/cos2θ)
Please help! Also if you have any tips on how to spot which integration techniques to use that would be great!
 
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  • #2
you might try some algebraic manipulation before you even try a trig sub...
 
  • #3
Ohhh! I can't believe I overlooked that, thanks!
 
  • #4
jdawg said:

Homework Statement


∫(x2+9x)/(4x2) dx

The sum is in the numerator, not the denominator, so the correct approach is to simplify.
 
  • #5
So now I have (1/4)∫(x+9)/(x) dx

u substitution doesn't work, I don't know what to do next.
 
  • #6
You still haven't quite taken mine and pasmith's advice yet. It turns out no substitution is necessary.
 
  • #7
Oh, maybe I didn't understand what you meant then. I thought you just meant to cancel out the x's?
 
  • #8
Write it as a sum of two integrals.
 
  • #9
Oh! Thanks so much, I forgot you could split up the numerator!
 

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