Answer:
[tex]\displaystyle \frac{1}{144}arctan(\frac{9x}{2}) + \frac{x}{8(81x^2 + 4)} + C[/tex]
General Formulas and Concepts:
Alg I
Terms/CoefficientsFactorExponential Rule [Dividing]: [tex]\displaystyle \frac{b^m}{b^n} = b^{m - n}[/tex]Pre-Calc
[Right Triangle Only] Pythagorean Theorem: a² + b² = c²
a is a legb is a legc is hypotenuseTrigonometric Ratio: [tex]\displaystyle sec(\theta) = \frac{1}{cos(\theta)}[/tex]
Trigonometric Identity: [tex]\displaystyle tan^2\theta + 1 = sec^2\theta[/tex]
TI: [tex]\displaystyle sin(2x) = 2sin(x)cos(x)[/tex]
TI: [tex]\displaystyle cos^2(\theta) = \frac{cos(2x) + 1}{2}[/tex]
Calc
Integration Rule [Reverse Power Rule]: [tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]
Integration Property [Multiplied Constant]: [tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]
IP [Addition/Subtraction]: [tex]\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx[/tex]
U-Substitution
U-Trig Substitution: x² + a² → x = atanθ
Step-by-step explanation:
Step 1: Define
[tex]\displaystyle \int {\frac{dx}{(81x^2 + 4)^2}}[/tex]
Step 2: Identify Sub Variables Pt.1
Rewrite integral [factor expression]:
[tex]\displaystyle \int {\frac{dx}{[(9x)^2 + 4]^2}}[/tex]
Identify u-trig sub:
[tex]\displaystyle x = atan\theta\\9x = 2tan\theta \rightarrow x = \frac{2}{9}tan\theta\\dx = \frac{2}{9}sec^2\theta d\theta[/tex]
Later, back-sub θ (integrate w/ respect to x):
[tex]\displaystyle tan\theta = \frac{9x}{2} \rightarrow \theta = arctan(\frac{9x}{2})[/tex]
Step 3: Integrate Pt.1
[Int] Sub u-trig variables: [tex]\displaystyle \int {\frac{\frac{2}{9}sec^2\theta}{[(2tan\theta)^2 + 4]^2}} \ d\theta[/tex][Int] Rewrite [Int Prop - MC]: [tex]\displaystyle \frac{2}{9} \int {\frac{sec^2\theta}{[(2tan\theta)^2 + 4]^2}} \ d\theta[/tex][Int] Evaluate exponents: [tex]\displaystyle \frac{2}{9} \int {\frac{sec^2\theta}{[4tan^2\theta + 4]^2}} \ d\theta[/tex][Int] Factor: [tex]\displaystyle \frac{2}{9} \int {\frac{sec^2\theta}{[4(tan^2\theta + 1)]^2}} \ d\theta[/tex][Int] Rewrite [TI]: [tex]\displaystyle \frac{2}{9} \int {\frac{sec^2\theta}{[4sec^2\theta]^2}} \ d\theta[/tex][Int] Evaluate exponents: [tex]\displaystyle \frac{2}{9} \int {\frac{sec^2\theta}{16sec^4\theta} \ d\theta[/tex][Int] Rewrite [Int Prop - MC]: [tex]\displaystyle \frac{1}{72} \int {\frac{sec^2\theta}{sec^4\theta} \ d\theta[/tex][Int] Divide [ER - D]: [tex]\displaystyle \frac{1}{72} \int {\frac{1}{sec^2\theta} \ d\theta[/tex][Int] Rewrite [TR]: [tex]\displaystyle \frac{1}{72} \int {cos^2\theta} \ d\theta[/tex][Int] Rewrite [TI]: [tex]\displaystyle \frac{1}{72} \int {\frac{cos(2\theta) + 1}{2}} \ d\theta[/tex][Int] Rewrite [Int Prop - MC]: [tex]\displaystyle \frac{1}{144} \int {cos(2\theta) + 1} \ d\theta[/tex][Int] Rewrite [Int Prop - A/S]: [tex]\displaystyle \frac{1}{144} [\int {cos(2\theta) \ d\theta + \int {1} \ d\theta][/tex]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]: [tex]\displaystyle \frac{1}{144} [\frac{1}{2} \int {2cos(2\theta) \ d\theta + \int {1 \theta ^0} \ d\theta][/tex][Int] U-Sub: [tex]\displaystyle \frac{1}{144} [\frac{1}{2} \int {cos(u) \ du + \int {1 \theta ^0} \ d\theta][/tex][Ints] Integrate [Trig/Int Rule - RPR]: [tex]\displaystyle \frac{1}{144} [\frac{1}{2} sin(u) + \theta + C][/tex][Expression] Back Sub: [tex]\displaystyle \frac{1}{144} [\frac{1}{2} sin(2 \theta) + arctan(\frac{9x}{2}) + C][/tex][Exp] Rewrite [TI]: [tex]\displaystyle \frac{1}{144} [\frac{1}{2}(2sin(\theta)cos(\theta)) + arctan(\frac{9x}{2}) + C][/tex][Exp] Multiply: [tex]\displaystyle \frac{1}{144} [sin(\theta)cos(\theta) + arctan(\frac{9x}{2}) + C][/tex][Exp] Back Sub: [tex]\displaystyle \frac{1}{144} [sin(arctan(\frac{9x}{2}))cos(arctan(\frac{9x}{2})) + arctan(\frac{9x}{2}) + C][/tex]Step 6: Triangle
Find trig values:
[tex]\displaystyle tan\theta = \frac{9x}{2}[/tex]
[tex]\displaystyle \theta = arctan(\frac{9x}{2})[/tex]
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]: [tex]\displaystyle 2^2 + (9x)^2 = c^2[/tex]Evaluate exponents: [tex]\displaystyle 4 + 81x^2 = c^2[/tex][Equality Property] Square root both sides: [tex]\displaystyle \sqrt{4 + 81x^2} = c[/tex]Rewrite: [tex]c = \sqrt{81x^2 + 4}[/tex]Substitute into trig ratios:
[tex]\displaystyle sin\theta = \frac{9x}{\sqrt{81x^2 + 4}}[/tex]
[tex]\displaystyle cos\theta = \frac{2}{\sqrt{81x^2 + 4}}[/tex]
Step 7: Integrate Pt.3
[Exp] Sub variables [TR]: [tex]\displaystyle \frac{1}{144} [\frac{9x}{\sqrt{81x^2 + 4}} \cdot \frac{2}{\sqrt{81x^2 + 4}} + arctan(\frac{9x}{2}) + C][/tex][Exp] Multiply: [tex]\displaystyle \frac{1}{144} [\frac{18x}{81x^2 + 4} + arctan(\frac{9x}{2}) + C][/tex][Exp] Distribute: [tex]\displaystyle \frac{1}{144}arctan(\frac{9x}{2}) + \frac{x}{8(81x^2 + 4)} + C[/tex]The distance between two numbers on the number line is 15. If one of the numbers is 9, what are the two
possibilities for the other number?
smaller possible number =
larger possible number =
Answer:
The largest possible number is 24, the smallest possible number is -6.
Step-by-step explanation:
maths
Smaller possible number = -6 and larger possible number = 24.
What is an absolute value?
The absolute value of a number is the distance value from zero without considering the sign of a number. It is also represented as |x|, where x is any number.
Given:
The distance between two numbers on the number line is 15.
And the distance between the numbers is the absolute value.
And one of the numbers is 9.
Let x and -x are the numbers.
So, when the number is x,
distance,
x-9 = 15.
x = 24
And when the number is -x,
distance,
9 -(-x)= 15.
x = -6.
Therefore, the possible integers are -6 and 24.
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I ate water by drinking it how?
Answer:
There's no concern that water will dilute the digestive juices or interfere with digestion. In fact, drinking water during or after a meal actually aids digestion. Water is essential for good health. Water and other liquids help break down food so that your body can absorb the nutrients.
Answer:
chewing on it or by ice
Step-by-step explanation:
Evaluate. Please help me:)
Answer:
x = −5 /2
Step-by-step explanation:
Answer It With Solution!
Please Answer It Correct
A pail holds 6.3 liters of water. A bottle holds 3.5 liter less that water than the pail what is the volume of water in the bottle?
The correct answer is 2.8 liters
Explanation:
The volume of the water in the bottle can be calculated by subtracting 3.5 liters from the volume of water the pail can hold (6.3 liters) because the volume in the bottle is 3.5 liters less than the one in the pail. Below, you can find the process.
volume in the pail - 3.5 liters = volume in the bottle
6.3 liters - 3.5 liters = 2.8 liters
This means the volume of water in the bottle is 2.8 liters. Also, you can double-check this answer by adding the volume in the bottle and the 3.5 liters (2.8 + 3.5 = 6.3).
convert 1 4/5 into an improper fraction
Answer:
9/5
Step-by-step explanation:
I hope this helps im pretty sure its correct
square 1 is equal to
Answer:
1
Step-by-step explanation:
Answer:
4sides
Step-by-step explanation:
these is answer 4 sided
Fluffy the bunny ate 1/3 of a carrot in the morning and she ate some more at night by the end of the day she still had 1/5 of the carrot left to eat how much of the carrot did fluffy the bunny eat at night
Answer:
2/15
Step-by-step explanation:
first you have to make both numbers have an equal denominator the subtract how much that was left from how much she Initially ate
ASAP BE||CD. Find the value of x and AC.
I don’t understand
please answer :) will give brainlst
Answer:
Your answer would be B. A metal railing rusting in damp weather because anything that contains any sort of rust whats so ever is always considered a chemical change.
Surface Area of 20,70,90 Will give brainliest to correct answer
Answer:
19,000 cm
Step-by-step explanation:
Hope this helps
Answer:
19000 square unitsStep-by-step explanation:
It is assumed the rectangular prism with dimensions of 20, 70 and 90
Surface are is calculated by the formula:
SA = 2(lw + lh + wh)Substitute values and find the value:
SA = 2(20*70 + 20*90 + 70*90) = 19000 square units13. Given f(x) = X – 3 and g(x) = 4x, find f(g(x)).
A. 4x - 3
B. 4x - 12
C. 5x - 3
D. 3x - 3
Step-by-step explanation:
f(g(x))
= f(4x)
= (4x) - 3. (A)
Step-by-step explanation:
option A is the correct answer according to my opinion .
plz mark my answer as brainlist if you find it useful.
Which statement about the simplified binomial expansion of (a – b)n, where n is a positive integer, is true?
Answer:
A on Edge 2021 ;))
Step-by-step explanation:
All terms of the expansion are positive.
The true statement is all terms of the expansion are positive.
What are binomial expansions?Binomial expansions are expansions of expressions that have two terms
The expression is given as:
[tex](a - b)^n[/tex]
This means that the expression (a - b) is to be multiplied by itself, n times.
Hence, all terms of the expansion are positive.
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Help ?
Um answer this question
Answer:
144
Step-by-step explanation:
12 times 12= 144
Explain answer !!
( WILL GIVE BRAINLEST)
Answer:
Step-by-step explanation:
Larissa: y = 250 - 25x
Chucho: y = 30 +30x
4 weeks
A juice company currently sells juice in gallon-size containers but wants to diversify to provide quart-size containers. If the company produces the number of gallons of juice every quarter given in the table, how many quarts will be produced in the first quarter?
The company will produce quarts of juice in first quarter.
Answer:
Depends on how many gallojs but 4 quarts per gallon
Step-by-step explanation:
Answer:
it is 24
Step-by-step explanation:
I just know it cuz I did it. hope it helps u :)
Find the output, g, when the input, r, is 4.
g=25−3r
plzzzzzz help me i need to finish
Answer: g = 13 / Thats the output
Step-by-step explanation:
g = 25 - 3r
Its telling you the r is 4 since its the input.
g = 25 - 3 ( 4 ) see!! just plug in the number
Now Solve
g = 25 - 12
g = 13
Answer:
g = 13
Step-by-step explanation:
since r = 4, you make the equation :
g=25-3(4)
first you multiply 3 and 4 and you get 12
g=25-12
and 25 - 12 is 13 which equals :
g = 13
escribe los múltiplos de 7 menores que 50
Answer:
7, 14, 21, 28, 35, 42, 49
Step-by-step explanation:
7 × 1 = 7
7 × 2 = 14
7 × 3 = 21
7 × 4 = 28
7 × 5 = 35
7 × 6 = 42
7 × 7 = 49
The next multiple of 7 is 56, but this is over 50.
PLEASE HELP —>50 POINTS
What are the linear factors of 2x^4 +13x^3+27x^2+13x-15 if the remainder upon division with x^2 +4x +5 is zero?
Step by step solution :
STEP
1
:
Equation at the end of step 1
((((2•(x4))-(13•(x3)))+33x2)-13x)-15 = 0
STEP
2
:
Equation at the end of step
2
:
((((2•(x4))-13x3)+33x2)-13x)-15 = 0
STEP
3
:
Equation at the end of step
3
:
(((2x4 - 13x3) + 33x2) - 13x) - 15 = 0
STEP
4
:
Polynomial Roots Calculator :
4.1 Find roots (zeroes) of : F(x) = 2x4-13x3+27x2-13x-15
Polynomial Roots Calculator is a set of methods aimed at finding values of x for which F(x)=0
The linear factors of the given polynomial are (2x+3)(x+1)(x+2+i)(x+2-i)
What is linear factor?The linear factors of polynomials are the first-degree equations that act as the building block for the simple or complex higher-order polynomials.
Given that, a polynomial 2x⁴+13³+27x²+13x-15, the remainder upon division with x²+4x +5 is zero,
That means, x²+4x +5 is a factor of the given polynomial,
On diving 2x⁴+13³+27x²+13x-15 by x²+4x +5 we get 2x²+5x+3 as quotient that means, x²+4x +5 and 2x²+5x+3 are the factors of the polynomial,
On factorizing both the factors we get,
(2x+3)(x+1)(x+2+i)(x+2-i)
Hence, the linear factors of the given polynomial are (2x+3)(x+1)(x+2+i)(x+2-i)
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Find The square root of -25 that graphs on the positive Y axis , please help
Answer:
5 and 5 because the squre root of 5 is 25
Please hurry; Rajiv bought a rare stamp for $125. A function that models the value of Rajiv’s after t years is v(t) = 125 (1.05)t . What is the value of Rajiv’s stamp after 20 years?
a $131.25
b $331.66
c $2,625.00
d $3316.62
Answer:
C.
Step-by-step explanation:
On the first day of a social media ad campaign, a website had 15 unique visitors. Over the following several days, the website received two times the number of unique visitors as the previous day.
Which equation can be used to determine the number of unique visitors, v, that the website would have on day d of the ad campaign?
Please only reply if you know the answer not just to get points I really need help
A) v= 15x2^d
B) v=2x15^d
C) v=2x15^(d-1)
D) v=15x2^(d-1)
Answer:
the correct answer is d :)
Step-by-step explanation:
the ratio of doors to windows is 1:5 . There are 186 doors and windows in a building . HOw many doors and windows are there?
Answer:
a. Doors = 31 doors
b. Windows =155 windows
Step-by-step explanation:
Given the following data;
Ratio D:W = 1:5 = 1 + 5 = 6
Total number of doors and windows = 186
Ratio of doors to windows = 1/6
Ratio of windows to doors = 5/6
To find the ratio of the doors;
Doors = 1/6 * 186
Doors = 186/6
Doors =31 doors
To find the ratio of windows;
Windows = 5/6 * 186
Windows = 930/6
Windows =155 windows
What is 4000 divided by 70
Answer:
57.1428571429
Step-by-step explanation:
The proper calculation of 4000 divided by 70 is approximately 57.14285714285714.
To calculate 4000 divided by 70 properly, we can perform the division:
4000 ÷ 70 = 57.14285714285714
The quotient of 4000 divided by 70 is approximately 57.14285714285714.
When dividing, 70 can be divided into 4000 approximately 57 times, with a remainder. To obtain a more precise decimal representation, the division can be carried out to more decimal places. However, for the purpose of this calculation, we have rounded the result to 14 decimal places.
Therefore, the proper calculation of 4000 divided by 70 is approximately 57.14285714285714.
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5/6x+2/3x+4=7
What is x and I REALLY REALLY NEED AN EXPLANATION
10
Bridget was thinking of a number. Bridget adds 8 to it, then doubles it and gets an answer of 21.9. What was the original number?
Answer: Bridget was thinking of the number 2.95
Step-by-step explanation:
21.9 / 2 = 10.95
10.95 - 8 = 2.95
Is this expression equivalent to 1.5?
1.5 / 10
(1.5 divided by 100)
Answer:
Very False
Step-by-step explanation:
1.5 divided by 100 I can tell you is not equivalent to 1.5
Owls eat voles (a relative of mice); voles eat grass and seeds. Which food chain is correct?
grass ----> Vole ----> Owl
Hope this helped :3
5x + 3x = 5x + 6 What value of 3x makes the equation true?
Answer:
-45 (9x-20) - 3x = 45x-6
-405x+900-3x = 45x-6
-408x+900 = 45x-6 (combine like terms)
906=453x (add 6 to 900 and - 408x to 45x)
2=x (divided )
X=2
Step-by-step explanation: I hope this helps.
what is 20:10 in its simplest
form
Answer:
2/1
Step-by-step explanation:
20/10 devided by 10 to both numbers
Intercepts from a graph
Determine the intercepts of the line.
y-intercept:
x-intercept:
y
0.4+
0.2
-1.4-1.2
0,2
-0.8 -0.6 -0.4-0.2
-0.21
0.6+
-0.81
1.2
Answer:
Y-intercept: (0,-0.7)
X-intercept: (-1.2,0)
Explanation:
The Y-intercept is the spot where the line crosses the Y axis (the vertical axis). This always occurs where X=0. For this problem the Y value at this point is -0.7. So, for the coordinates (X,Y) X is 0 and Y is -0.7 so it is (0,-0.7)
The X-intercept is the spot where the line crosses the X axis (the horizontal axis). This always occurs where Y=0. For this problem the X value at this point is -1.2. So, for the coordinates (X,Y) X is -1.2 and Y is 0 so it is (0,-0.7)