Mathematics, 10.02.2021 09:20 calebabaltimore

How do I evaluate this using trigonometric substitution?

∫dx/(81^2+4)^2

Answers: 1

Show answers

Answers

Answer from: Sauceyx7

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

General Formulas and Concepts:

Alg I

Terms/CoefficientsFactorExponential Rule [Dividing]: $\displaystyle \frac{b^m}{b^n} = b^{m - n}$

Pre-Calc

[Right Triangle Only] Pythagorean Theorem: a² + b² = c²

a is a legb is a legc 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: $\displaystyle cos^2(\theta) = \frac{cos(2x) + 1}{2}$

Calc

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:

Step 1: Define

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

Step 2: Identify Sub Variables Pt.1

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 x):

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

Step 3: Integrate Pt.1

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

Step 4: Identify Sub Variables Pt.2

Determine u-sub for trig int:

u = 2θ

du = 2dθ

Step 5: Integrate Pt.2

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

Step 6: Triangle

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 a = 2

Leg b = 9x

Leg c = ?

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

Substitute into trig ratios:

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

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

Step 7: Integrate Pt.3

[Exp] Sub variables [TR]: $\displaystyle \frac{1}{144} [\frac{9x}{\sqrt{81x^2 + 4}} \cdot \frac{2}{\sqrt{81x^2 + 4}} + arctan(\frac{9x}{2}) + C]$ [Exp] Multiply: $\displaystyle \frac{1}{144} [\frac{18x}{81x^2 + 4} + arctan(\frac{9x}{2}) + C]$ [Exp] Distribute: $\displaystyle \frac{1}{144}arctan(\frac{9x}{2}) + \frac{x}{8(81x^2 + 4)} + C$

Answer from: Quest

two data sets with the same median will not have the same varaibility

step-by-step explanation:

Answer from: Quest

the answer is thin because within context, the stanza concludes the and frail tone the people have toward each other, and the fact that it would be thin, makes the most sense compared to "pliable" or "not hard or firm to the touch". hope this , good luck.

step-by-step explanation:

can i have brainliest

Another question on Mathematics

Mathematics, 21.06.2019 16:10

Which of the following graphs represents the function f(x) = 2^x

Answers: 2

Answer

Mathematics, 21.06.2019 20:50

Including 6% sales tax, an inn charges $135.68 per night. find the inns nightly cost

Answers: 1

Answer

Mathematics, 21.06.2019 21:30

Iwill give brainliest. suppose tommy walks from his home at (0, 0) to the mall at (0, 5), and then walks to a movie theater at (6, 5). after leaving the theater tommy walks to the store at (6, 0) before returning home. if each grid square represents one block, how many blocks does he walk?

Answers: 2

Answer

Mathematics, 21.06.2019 21:50

Which value of y will make the inequality y< -1 false?

Answers: 2

Answer

You know the right answer?

How do I evaluate this using trigonometric substitution?

∫dx/(81^2+4)^2...

Questions