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Bumek [7]
2 years ago
9

How do I evaluate this using trigonometric substitution?∫dx/(81x^2+4)^2

Mathematics
1 answer:
Daniel [21]2 years ago
3 0

Answer:

\displaystyle \frac{1}{144}arctan(\frac{9x}{2}) + \frac{x}{8(81x^2 + 4)} + C

General Formulas and Concepts:

<u>Alg I</u>

  • Terms/Coefficients
  • Factor
  • Exponential Rule [Dividing]: \displaystyle \frac{b^m}{b^n} = b^{m - n}

<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: \displaystyle sec(\theta) = \frac{1}{cos(\theta)}

Trigonometric Identity: \displaystyle tan^2\theta + 1 = sec^2\theta

TI: \displaystyle sin(2x) = 2sin(x)cos(x)

TI: \displaystyle cos^2(\theta) = \frac{cos(2x) + 1}{2}

<u>Calc</u>

Integration Rule [Reverse Power Rule]:                                                                \displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C

Integration Property [Multiplied Constant]:                                                         \displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx

IP [Addition/Subtraction]:                                                             \displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx

U-Substitution

U-Trig Substitution: x² + a² → x = atanθ

Step-by-step explanation:

<u>Step 1: Define</u>

\displaystyle \int {\frac{dx}{(81x^2 + 4)^2}}

<u>Step 2: Identify Sub Variables Pt.1</u>

Rewrite integral [factor expression]:

\displaystyle \int {\frac{dx}{[(9x)^2 + 4]^2}}

Identify u-trig sub:

\displaystyle x = atan\theta\\9x = 2tan\theta \rightarrow x = \frac{2}{9}tan\theta\\dx = \frac{2}{9}sec^2\theta d\theta

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

\displaystyle tan\theta = \frac{9x}{2}  \rightarrow \theta = arctan(\frac{9x}{2})

<u>Step 3: Integrate Pt.1</u>

  1. [Int] Sub u-trig variables:                                                                                 \displaystyle \int {\frac{\frac{2}{9}sec^2\theta}{[(2tan\theta)^2 + 4]^2}} \ d\theta
  2. [Int] Rewrite [Int Prop - MC]:                                                                           \displaystyle \frac{2}{9} \int {\frac{sec^2\theta}{[(2tan\theta)^2 + 4]^2}} \ d\theta
  3. [Int] Evaluate exponents:                                                                                \displaystyle \frac{2}{9} \int {\frac{sec^2\theta}{[4tan^2\theta + 4]^2}} \ d\theta
  4. [Int] Factor:                                                                                                      \displaystyle \frac{2}{9} \int {\frac{sec^2\theta}{[4(tan^2\theta + 1)]^2}} \ d\theta
  5. [Int] Rewrite [TI]:                                                                                              \displaystyle \frac{2}{9} \int {\frac{sec^2\theta}{[4sec^2\theta]^2}} \ d\theta
  6. [Int] Evaluate exponents:                                                                                \displaystyle \frac{2}{9} \int {\frac{sec^2\theta}{16sec^4\theta} \ d\theta
  7. [Int] Rewrite [Int Prop - MC]:                                                                          \displaystyle \frac{1}{72} \int {\frac{sec^2\theta}{sec^4\theta} \ d\theta
  8. [Int] Divide [ER - D]:                                                                                         \displaystyle \frac{1}{72} \int {\frac{1}{sec^2\theta} \ d\theta
  9. [Int] Rewrite [TR]:                                                                                            \displaystyle \frac{1}{72} \int {cos^2\theta} \ d\theta
  10. [Int] Rewrite [TI]:                                                                                              \displaystyle \frac{1}{72} \int {\frac{cos(2\theta) + 1}{2}} \ d\theta
  11. [Int] Rewrite [Int Prop - MC]:                                                                          \displaystyle \frac{1}{144} \int {cos(2\theta) + 1} \ d\theta
  12. [Int] Rewrite [Int Prop - A/S]:                                                                          \displaystyle \frac{1}{144} [\int {cos(2\theta) \ d\theta + \int {1} \ d\theta]  

<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>

  1. [Ints] Rewrite [Int Prop - MC]:                                                                       \displaystyle \frac{1}{144} [\frac{1}{2} \int {2cos(2\theta) \ d\theta + \int {1 \theta ^0} \ d\theta]
  2. [Int] U-Sub:                                                                                                     \displaystyle \frac{1}{144} [\frac{1}{2} \int {cos(u) \ du + \int {1 \theta ^0} \ d\theta]
  3. [Ints] Integrate [Trig/Int Rule - RPR]:                                                             \displaystyle \frac{1}{144} [\frac{1}{2} sin(u) + \theta + C]
  4. [Expression] Back Sub:                                                                                 \displaystyle \frac{1}{144} [\frac{1}{2} sin(2 \theta) + arctan(\frac{9x}{2}) + C]
  5. [Exp] Rewrite [TI]:                                                                                           \displaystyle \frac{1}{144} [\frac{1}{2}(2sin(\theta)cos(\theta)) + arctan(\frac{9x}{2}) + C]
  6. [Exp] Multiply:                                                                                                 \displaystyle \frac{1}{144} [sin(\theta)cos(\theta) + arctan(\frac{9x}{2}) + C]
  7. [Exp] Back Sub:                                                                                             \displaystyle \frac{1}{144} [sin(arctan(\frac{9x}{2}))cos(arctan(\frac{9x}{2})) + arctan(\frac{9x}{2}) + C]

<u>Step 6: Triangle</u>

Find trig values:

\displaystyle tan\theta = \frac{9x}{2}

\displaystyle \theta = arctan(\frac{9x}{2})

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> = ?

  1. Sub variables [PT]:                                                                                         \displaystyle 2^2 + (9x)^2 = c^2
  2. Evaluate exponents:                                                                                      \displaystyle 4 + 81x^2 = c^2
  3. [Equality Property] Square root both sides:                                                  \displaystyle \sqrt{4 + 81x^2} = c
  4. Rewrite:                                                                                                           c = \sqrt{81x^2 + 4}

Substitute into trig ratios:

\displaystyle sin\theta = \frac{9x}{\sqrt{81x^2 + 4}}

\displaystyle cos\theta = \frac{2}{\sqrt{81x^2 + 4}}

<u>Step 7: Integrate Pt.3</u>

  1. [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]
  2. [Exp] Multiply:                                                                                                 \displaystyle \frac{1}{144} [\frac{18x}{81x^2 + 4} + arctan(\frac{9x}{2}) + C]
  3. [Exp] Distribute:                                                                                             \displaystyle \frac{1}{144}arctan(\frac{9x}{2}) + \frac{x}{8(81x^2 + 4)} + C
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