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]erica planted 8 cucumber plants each plant had 37 green cucumbers. how many cucumbers are the 8 plants altogether
Answer:
296
Step-by-step explanation:
8 plants × 37 on each = 296 cucumbers
Answer: 296
If you multiply the 37 green cucumbers each by the 8 cucumber plants they grew from, you'd end up with 296 green cucumbers altogether.
How much water is in a empty glass that is 10 centimeters high and has a diameter of 5 cm
what is the square root of 36
Answer:
6 !
Step-by-step explanation:
since 6 x 6 = 36, the square root is 6 !
square root is when u multiply two same numbers for the output... so like 49 would b 7, since 7 x 7 = 49 !!
im not sure if that makes any sense, but i hope it helps !!
Write an equation for the function whose graph is described.
The shape of f(x) = |x|, but shifted 19 units up and then reflected in the x-axis
Answer:
f(x)=-|x|+19
Step-by-step explanation:
so shifted 19 units up means
abosolute value of x + 19
and reflected in the x axis means that you have to put the - before the abosolute value.
so -|x|+19
Harvey the wonder hamster can run 3 1/6 km in 1/4 hour. Harvey runs at a constant rate
Find his average speed in kilometers per hour.
Answer:
12 2/3 KMPH
Step-by-step explanation:
3 1/6 x 4 (reciprical of 1/4) = 12 2/3
the cost of New York Yankees baseball cap is 24.00. The local sporting goods store sells it for $30.00. markup rate
PLEASE HELP IN ONE MINUTE WILL MARK BRAINLIST
Answer:
1,-2 | -2,7 ????
Step-by-step explanation:
Increase 480 by 17%
Quick answer please!!
Answer:
561.6
Step-by-step explanation:
480+17%(81.6)=561.6
Answer:
561.6
Step-by-step explanation:
Percentage increase = 17% × 480
= 81.6
New value = 480 + 81.6
= 561.6
One number is seven more than six times another number. If the sum of the numbers is 21, find the numbers
Answer:the two numbers are 2 and 19
Step-by-step explanation:
Let one of the numbers be x
and the second be 7+ 6x
so that the sum of the two numbers which totals 21 be expressed as
x + (7+ 6x) = 21
x+ 7+6x= 21
7 + 7x = 21
7x= 14
x= 14/7 = 2
one of the numbers = 2
the other number given as
7+ 6x = 7+ 6 x 2= 7+12= 19
The numbers are 2 and 19 .
The numbers as described in the question are; 2 and 19
Solving word problemsFrom the given problem statements;
The first number = y
The second number = x
Hence, y = (7 + 6x)
Their sum is therefore;
x + 7 = 6x = 217x = 14x = 2Hence, the second number is; 7 + 6x = 7 + 6(2);
y = 19Read more on word problems;
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Jeffrey has an aquarium in the shape of a rectangular prism. He is going to place a layer of sand over the base for his iguana. The area of the base of the aquarium can be represented by the expression (42x3−14x2+21x) feet3. If the width of the aquarium is represented by the expression (7x) feet, what is the length of the aquarium?
Answer:
[tex]Length = 6x^2 - 2x + 3[/tex]
Step-by-step explanation:
Given
[tex]Area = 42x^3 - 14x^2 + 21x[/tex]
[tex]Width = 7x[/tex]
Required
Determine the length of the aquarium
The area of a rectangular base is calculated as:
[tex]Area = Length * Width[/tex]
Substitute values for Area and Width
[tex]42x^3 - 14x^2 + 21x = Length * 7x[/tex]
Make Length the subject
[tex]Length = \frac{42x^3 - 14x^2 + 21x}{7x}[/tex]
Factorize the numerator
[tex]Length = \frac{7x(6x^2 - 2x + 3)}{7x}[/tex]
[tex]Length = 6x^2 - 2x + 3[/tex]
The tape diagram represents an equation. 7 Write an equation to represent the image. + Report a problem
Step-by-step explanation:
y+y = 7
.................
How old am I if, eighty reduced by 2 times my age is fifty-six
The principal at a school recorded the color and make of all vehicles in the student parking lot. From the data, the following probability model was created about the colors of the vehicles.
A 2-column table with 4 rows. Column 1 is labeled outcome with entries red, black, white, other. Column 2 is labeled probability with entries 0.28, 0.30, 0.22, 0.20.
Which pairs of events about a vehicle are mutually exclusive? Check all that apply.
The events red and black
The events white and two door
The events white and other
The events black and truck
The events red and other
Answer:
1, 3, 5
Step-by-step explanation:
EDG 2020-2021 Stats
The events ,The events red and black ,The events white and other, The events red and other are mutually exclusive.
What is probability?The probability is defined as the chances of happening any event. Mutually exclusive events are those in which the one event is independent of the other event.
The condition for the mutually exclusive event will be:
[tex]P(A|B)=\dfrac{P(A) \ and P(B)}{P(A)}=P(A)[/tex]
We have the data
red, 0.28
black, 0.30
white, 0.22
other 0.20
Here we will check the mutually exclusive event:-
(1)The events red and black
[tex]P(R|B)=\dfrac{P(R\ and\ B)}{P(B)}[/tex]
[tex]P(R|B)=\dfrac{0.28\times 0.30}{0.30}=0.28 =P(R)[/tex]
Hence the event is exclusive.
(2)The events white and other
[tex]P(W|O)=\dfrac{P(W\ and\ O)}{P(O)}=P(W)[/tex]
[tex]P(W|O)=\dfrac{ 0.22\times 0.20}{0.20}=0.22=P(W)[/tex]
Hence the event is exclusive.
(3) The events red and other
[tex]P(R|O)=\dfrac{P(R\ and\ O)}{P(O)}=P(R)[/tex]
[tex]P(R|O)=\dfrac{ 0.28\times 0.20}{0.20}=0.28=P(R)[/tex]
Hence the event is exclusive.
Hence the events ,The events red and black ,The events white and other, The events red and other are mutually exclusive.
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At this weekend’s football game, 15 out of the first 50 people who entered the field were not wearing hats. If this sample is representative of the 250 people attending the game, about how many of them will probably NOT be wearing hats?
75 people out of 250 will NOT be wearing hats.
At this weekend's football game, 15 out of the first 50 people entered in the field were not wearing hats.
The number of people attending the game is 250.
What is Probability?
Probability is a measure of the likelihood of an event to occur.
Now,
Probability for not wearing hats;
⇒ 250/ 50 = 5
⇒ 15 x 5 = 75
Hence, 75 people out of 250 will NOT be wearing hats.
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The new Nintendo game costs $70. The sales tax was 5%, what was the total cost?
Answer:
$66.50.
Step-by-step explanation:
Mom is planting a rose garden in the backyard. The scale drawing shows the garden. What is the actual width and length of the rose garden?
A: length = 34 feet
B: width = 10 feet
C: width = 5 feet
D: length = 17 feet
Answer:
34
Step-by-step explanation:
Find the 18h term of the arithmetic sequence whose common difference is d=4 and whose first term is a, =3.
Answer:
the 18th term is 71
Step-by-step explanation:
The general form for any term "n" in an arithmetic sequence is:
a(n) = a1 + (n-1) * d
where "d" is the common difference.
So, in our case we have:
a(n) = 3 + (n-1) *4
Therefore, the 18th term will be:
a(18) = 3 + (18 - 1) 4 = 3 + 17 * 4 = 71
The perimeter of a rectangular carpet is 94 yards. Its width is 19 yards.
How long is the carpet?
label required
Answer:
28 yards
Step-by-step explanation:
Perimeter=2(L)+2(W)
94=2(L)+2(19)
94=2(L)+38
-38 -38
56=2(L)
÷2 ÷2
28=L
I need help with math
Answer:
6:3
Step-by-step explanation:
A poll is taken to see which candidate students will elect as class president. 43% of
students say they will vote for Alessandra. The poll has a margin of error of 3%, so the
least and greatest percent she could receive is represented by la – 431 = 3, where a is
the actual percent of students that vote for Alessandra. What is the lowest percentage of
votes she could receive?
Answer:
The lowest percentage of votes she could receive is 40%
Step-by-step explanation:
43% of students say they will vote for Alessandra, with a margin of error of 3%.
This means that between 43% - 3% = 40% and 43% + 3% = 46% of the students will vote for Alessandra.
This means that the lowest percentage of votes she could receive is 40% and the highest is 46%.
1/2 (16x + 20)
Need help
Answer: let's say x = 1
16 x 1 =16
16+20=36
36x1/2=18
Step-by-step explanation:
Find the median of these numbers:4,2,7,4,3 *
Answer:
4
Step-by-step explanation:
median: the middle value. to find this, arrange your numbers in order from least to greatest and find the one in the middle. if there are two in the middle, add them and then divide by 2.
4, 2, 7, 4, 3 → 2, 3, 4, 4, 7
since 4 is in the middle, that is your answer
The median of the sequence is 4.
What is the median?One of the three methods of measuring central tendency is the median. The centre point of a data set is indicated while describing it. The measure of central tendency is this. The mean, median, and mode are the three most popular methods for calculating central tendency.
Given the sequence 4,2,7,4,3,
to find median arrange the values in ascending order,
2, 3, 4, 4, 7
since the sequence has 5 digits so the median is the middle term,
center digit is 4,
median is 4.
Hence the median is 4
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diego has some money in his bank account before he starts a summer job. he deposits the money from the summer job in his bank account and doesn't spend any of it. After working 3 hours, he has $71 in the account. After working 12 hours, he has $134 in the account.
How much money does diego earn per hour
Answer:
About $23.77 per hour.
Step-by-step explanation:
Just thank me later and click the crown at the top of this answer if this helped ;)
The Diego earn [tex]\$ 7[/tex] per hour.
Calculation:Given that,
After working 3 hours, he has $71 in the account.
After working 12 hours, he has $134 in the account.
It means that, After working 9 hours,
Diego earn [tex]=134-71=63[/tex]
In 9 hours, earning [tex]=63\$[/tex]
In one hour, earning [tex]=63/9=\$7[/tex]
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What is the slope-intercept equation of this line? (0.6) (4.-2)
Answer: y=-2x+6
Step-by-step explanation:
The equation for slope-intercept form is : y=mx+b
m=slopeb=y-intercept----------------------------------------------------------------
First Step: find the slope
- (y2-y1)/(x2-x1)
GIven
x1=0
y1=6
x2=4
y2=-2
Solve
(y2-y1)/(x2-x1)
=(-2-6)/(4-0)
=-8/4
=-2
Second Step: find the y-intercept
Given
y=-2x+b
Substitute one of the points into the equation
y=-2x+b
(-2)=-2(4)+b
-2=-8+b
b=6
Hope this helps!! :)
Please let me know if you have any questions
It’s due late tonight can some pls help me pass
Can you help me on question one?!
Answer:
I believe the answer is C
Answer:
I think answer is C
Step-by-step explanation:
Becausr A B AND D ARE SAME FORM OF WRITTING
Which is greater. 12 laps in 8 minutes or 16 laps in 10 minutes
Answer:
16 laps in one minute
Step-by-step explanation:
If you divide it you can find that the person will run 1.6 laps in a minute and the other one will run 1.5 laps in 1 minute.
what is 900 * 158000
Answer:
142,200,000
Step-by-step explanation:
Used Calculator
Answer:
142200000
Step-by-step explanation:
It's a really long answer
There are 3 space helmets and you take away 2,how many do you have?
Answer:
1 space helmet
Step-by-step explanation:
subtract 3-2 = 1 helmet
A rental car company charges a fee of $150.00 for renting a car for a week plus $0.25 per mile driven.
Answer:
y = 150z + .25x
y = $ amount of rental
z = weeks rented
x = miles driven
i hope this helps :)
