What type of integration to use?

[jdawg]

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!

 

[Pythagorean]

you might try some algebraic manipulation before you even try a trig sub...

 

[jdawg]

Ohhh! I can't believe I overlooked that, thanks!

 

[pasmith]

Homework Statement

∫(x²+9x)/(4x²) dx

The sum is in the numerator, not the denominator, so the correct approach is to simplify.

 

[jdawg]

So now I have (1/4)∫(x+9)/(x) dx

u substitution doesn't work, I don't know what to do next.

 

[Pythagorean]

You still haven't quite taken mine and pasmith's advice yet. It turns out no substitution is necessary.

 

[jdawg]

Oh, maybe I didn't understand what you meant then. I thought you just meant to cancel out the x's?

 

[D H]

Write it as a sum of two integrals.

 

[jdawg]

Oh! Thanks so much, I forgot you could split up the numerator!