Answer:

General Formulas and Concepts:
<u>Alg I</u>
- Terms/Coefficients
- Factor
- Exponential Rule [Dividing]:

<u>Pre-Calc</u>
[Right Triangle Only] Pythagorean Theorem: a² + b² = c²
- a is a leg
- b is a leg
- c is hypotenuse
Trigonometric Ratio: 
Trigonometric Identity: 
TI: 
TI: 
<u>Calc</u>
Integration Rule [Reverse Power Rule]: 
Integration Property [Multiplied Constant]: 
IP [Addition/Subtraction]: ![\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%7B%5Bf%28x%29%20%5Cpm%20g%28x%29%5D%7D%20%5C%2C%20dx%20%3D%20%5Cint%20%7Bf%28x%29%7D%20%5C%2C%20dx%20%5Cpm%20%5Cint%20%7Bg%28x%29%7D%20%5C%2C%20dx)
U-Substitution
U-Trig Substitution: x² + a² → x = atanθ
Step-by-step explanation:
<u>Step 1: Define</u>

<u>Step 2: Identify Sub Variables Pt.1</u>
Rewrite integral [factor expression]:
![\displaystyle \int {\frac{dx}{[(9x)^2 + 4]^2}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%7B%5Cfrac%7Bdx%7D%7B%5B%289x%29%5E2%20%2B%204%5D%5E2%7D%7D)
Identify u-trig sub:

Later, back-sub θ (integrate w/ respect to <em>x</em>):

<u>Step 3: Integrate Pt.1</u>
- [Int] Sub u-trig variables:
![\displaystyle \int {\frac{\frac{2}{9}sec^2\theta}{[(2tan\theta)^2 + 4]^2}} \ d\theta](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%7B%5Cfrac%7B%5Cfrac%7B2%7D%7B9%7Dsec%5E2%5Ctheta%7D%7B%5B%282tan%5Ctheta%29%5E2%20%2B%204%5D%5E2%7D%7D%20%5C%20d%5Ctheta)
- [Int] Rewrite [Int Prop - MC]:
![\displaystyle \frac{2}{9} \int {\frac{sec^2\theta}{[(2tan\theta)^2 + 4]^2}} \ d\theta](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B2%7D%7B9%7D%20%5Cint%20%7B%5Cfrac%7Bsec%5E2%5Ctheta%7D%7B%5B%282tan%5Ctheta%29%5E2%20%2B%204%5D%5E2%7D%7D%20%5C%20d%5Ctheta)
- [Int] Evaluate exponents:
![\displaystyle \frac{2}{9} \int {\frac{sec^2\theta}{[4tan^2\theta + 4]^2}} \ d\theta](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B2%7D%7B9%7D%20%5Cint%20%7B%5Cfrac%7Bsec%5E2%5Ctheta%7D%7B%5B4tan%5E2%5Ctheta%20%2B%204%5D%5E2%7D%7D%20%5C%20d%5Ctheta)
- [Int] Factor:
![\displaystyle \frac{2}{9} \int {\frac{sec^2\theta}{[4(tan^2\theta + 1)]^2}} \ d\theta](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B2%7D%7B9%7D%20%5Cint%20%7B%5Cfrac%7Bsec%5E2%5Ctheta%7D%7B%5B4%28tan%5E2%5Ctheta%20%2B%201%29%5D%5E2%7D%7D%20%5C%20d%5Ctheta)
- [Int] Rewrite [TI]:
![\displaystyle \frac{2}{9} \int {\frac{sec^2\theta}{[4sec^2\theta]^2}} \ d\theta](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B2%7D%7B9%7D%20%5Cint%20%7B%5Cfrac%7Bsec%5E2%5Ctheta%7D%7B%5B4sec%5E2%5Ctheta%5D%5E2%7D%7D%20%5C%20d%5Ctheta)
- [Int] Evaluate exponents:

- [Int] Rewrite [Int Prop - MC]:

- [Int] Divide [ER - D]:

- [Int] Rewrite [TR]:

- [Int] Rewrite [TI]:

- [Int] Rewrite [Int Prop - MC]:

- [Int] Rewrite [Int Prop - A/S]:
<u>Step 4: Identify Sub Variables Pt.2</u>
Determine u-sub for trig int:
u = 2θ
du = 2dθ
<u>Step 5: Integrate Pt.2</u>
- [Ints] Rewrite [Int Prop - MC]:
![\displaystyle \frac{1}{144} [\frac{1}{2} \int {2cos(2\theta) \ d\theta + \int {1 \theta ^0} \ d\theta]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B1%7D%7B144%7D%20%5B%5Cfrac%7B1%7D%7B2%7D%20%5Cint%20%7B2cos%282%5Ctheta%29%20%5C%20d%5Ctheta%20%2B%20%5Cint%20%7B1%20%5Ctheta%20%5E0%7D%20%5C%20d%5Ctheta%5D)
- [Int] U-Sub:
![\displaystyle \frac{1}{144} [\frac{1}{2} \int {cos(u) \ du + \int {1 \theta ^0} \ d\theta]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B1%7D%7B144%7D%20%5B%5Cfrac%7B1%7D%7B2%7D%20%5Cint%20%7Bcos%28u%29%20%5C%20du%20%2B%20%5Cint%20%7B1%20%5Ctheta%20%5E0%7D%20%5C%20d%5Ctheta%5D)
- [Ints] Integrate [Trig/Int Rule - RPR]:
![\displaystyle \frac{1}{144} [\frac{1}{2} sin(u) + \theta + C]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B1%7D%7B144%7D%20%5B%5Cfrac%7B1%7D%7B2%7D%20sin%28u%29%20%2B%20%5Ctheta%20%2B%20C%5D)
- [Expression] Back Sub:
![\displaystyle \frac{1}{144} [\frac{1}{2} sin(2 \theta) + arctan(\frac{9x}{2}) + C]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B1%7D%7B144%7D%20%5B%5Cfrac%7B1%7D%7B2%7D%20sin%282%20%5Ctheta%29%20%2B%20arctan%28%5Cfrac%7B9x%7D%7B2%7D%29%20%2B%20C%5D)
- [Exp] Rewrite [TI]:
![\displaystyle \frac{1}{144} [\frac{1}{2}(2sin(\theta)cos(\theta)) + arctan(\frac{9x}{2}) + C]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B1%7D%7B144%7D%20%5B%5Cfrac%7B1%7D%7B2%7D%282sin%28%5Ctheta%29cos%28%5Ctheta%29%29%20%2B%20arctan%28%5Cfrac%7B9x%7D%7B2%7D%29%20%2B%20C%5D)
- [Exp] Multiply:
![\displaystyle \frac{1}{144} [sin(\theta)cos(\theta) + arctan(\frac{9x}{2}) + C]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B1%7D%7B144%7D%20%5Bsin%28%5Ctheta%29cos%28%5Ctheta%29%20%2B%20arctan%28%5Cfrac%7B9x%7D%7B2%7D%29%20%2B%20C%5D)
- [Exp] Back Sub:
![\displaystyle \frac{1}{144} [sin(arctan(\frac{9x}{2}))cos(arctan(\frac{9x}{2})) + arctan(\frac{9x}{2}) + C]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B1%7D%7B144%7D%20%5Bsin%28arctan%28%5Cfrac%7B9x%7D%7B2%7D%29%29cos%28arctan%28%5Cfrac%7B9x%7D%7B2%7D%29%29%20%2B%20arctan%28%5Cfrac%7B9x%7D%7B2%7D%29%20%2B%20C%5D)
<u>Step 6: Triangle</u>
Find trig values:


tanθ = opposite / adjacent; solve hypotenuse of right triangle, determine trig ratios:
sinθ = opposite / hypotenuse
cosθ = adjacent / hypotenuse
Leg <em>a</em> = 2
Leg <em>b</em> = 9x
Leg <em>c</em> = ?
- Sub variables [PT]:

- Evaluate exponents:

- [Equality Property] Square root both sides:

- Rewrite:

Substitute into trig ratios:


<u>Step 7: Integrate Pt.3</u>
- [Exp] Sub variables [TR]:
![\displaystyle \frac{1}{144} [\frac{9x}{\sqrt{81x^2 + 4}} \cdot \frac{2}{\sqrt{81x^2 + 4}} + arctan(\frac{9x}{2}) + C]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B1%7D%7B144%7D%20%5B%5Cfrac%7B9x%7D%7B%5Csqrt%7B81x%5E2%20%2B%204%7D%7D%20%5Ccdot%20%5Cfrac%7B2%7D%7B%5Csqrt%7B81x%5E2%20%2B%204%7D%7D%20%2B%20arctan%28%5Cfrac%7B9x%7D%7B2%7D%29%20%2B%20C%5D)
- [Exp] Multiply:
![\displaystyle \frac{1}{144} [\frac{18x}{81x^2 + 4} + arctan(\frac{9x}{2}) + C]](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B1%7D%7B144%7D%20%5B%5Cfrac%7B18x%7D%7B81x%5E2%20%2B%204%7D%20%2B%20arctan%28%5Cfrac%7B9x%7D%7B2%7D%29%20%2B%20C%5D)
- [Exp] Distribute:
