The indefinite integral of ((6e^4x)(e^arctan(3e^4x)))/(1+9e^8x)
Answers
Answer:
[tex]\displaystyle \int {\bigg( 6e^\big{4x} \cdot \frac{e^\big{arctan(3e^{4x})}}{1 + 9e^\big{8x}} \bigg)} \, dx = \frac{e^\big{arctan(3e^{4x})}}{2} + C[/tex]
General Formulas and Concepts:
Calculus
Differentiation
DerivativesDerivative NotationDerivative Property [Multiplied Constant]: [tex]\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)[/tex]
Basic Power Rule:
f(x) = cxⁿf’(x) = c·nxⁿ⁻¹Derivative Rule [Chain Rule]: [tex]\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]
Integration
Integrals[Indefinite Integrals] Integration Constant CIntegration Property [Multiplied Constant]: [tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]
U-Substitution
U-SolveStep-by-step explanation:
Step 1: Define
Identify
[tex]\displaystyle \int {\bigg( 6e^\big{4x} \cdot \frac{e^\big{arctan(3e^{4x})}}{1 + 9e^\big{8x}} \bigg)} \, dx[/tex]
Step 2: Integrate Pt. 1
[Integrand] Rewrite: [tex]\displaystyle \int {\bigg( 6e^\big{4x} \cdot \frac{e^\big{arctan(3e^{4x})}}{1 + 9e^\big{8x}} \bigg)} \, dx = \int {\frac{6e^\big{arctan(3e^{4x}) + 4x}}{1 + 9e^\big{8x}}} \, dx[/tex][Integral] Rewrite [Integration Property - Multiplied Constant]: [tex]\displaystyle \int {\bigg( 6e^\big{4x} \cdot \frac{e^\big{arctan(3e^{4x})}}{1 + 9e^\big{8x}} \bigg)} \, dx = 6\int {\frac{e^\big{arctan(3e^{4x}) + 4x}}{1 + 9e^\big{8x}}} \, dx[/tex]Step 3: integrate Pt. 2
Set variables for u-substitution.
Set u: [tex]\displaystyle u = 4x[/tex][u] Differentiate [Basic Power Rule, Multiplied Constant]: [tex]\displaystyle du = 4 \ dx[/tex]Step 4: Integrate Pt. 3
[Integral] Rewrite [Integration Property - Multiplied Constant]: [tex]\displaystyle \int {\bigg( 6e^\big{4x} \cdot \frac{e^\big{arctan(3e^{4x})}}{1 + 9e^\big{8x}} \bigg)} \, dx = \frac{3}{2}\int {\frac{4e^\big{arctan(3e^{4x}) + 4x}}{1 + 9e^\big{8x}}} \, dx[/tex][Integral] U-Substitution: [tex]\displaystyle \int {\bigg( 6e^\big{4x} \cdot \frac{e^\big{arctan(3e^{4x})}}{1 + 9e^\big{8x}} \bigg)} \, dx = \frac{3}{2}\int {\frac{e^\big{arctan(3e^u) + u}}{1 + 9e^\big{2u}}} \, du[/tex]Step 5: Integrate Pt. 4
Set variables for u-substitution #2.
Set v: [tex]\displaystyle v = 9e^{2u} + 1[/tex] [v] Differentiate [Exponential Differentiation, Chain Rule]: [tex]\displaystyle dv = 18e^{2u} \ du[/tex][v] U-Solve: [tex]\displaystyle u = ln \Big( \frac{\sqrt{v - 1}}{3} \Big)[/tex][dv] U-Solve: [tex]\displaystyle du = \frac{e^{-2u}}{18} \ dv[/tex][U-Solve] Rewrite u: [tex]\displaystyle e^u = \frac{\sqrt{v - 1}}{3}[/tex]Step 6: Integrate Pt. 5
[Integral] U-Solve: [tex]\displaystyle \int {\bigg( 6e^\big{4x} \cdot \frac{e^\big{arctan(3e^{4x})}}{1 + 9e^\big{8x}} \bigg)} \, dx = \frac{3}{2}\int {\frac{e^\big{arctan(3(\frac{\sqrt{v - 1}}{3})) + ln(\frac{\sqrt{v - 1}}{3})}}{1 + 9(\frac{\sqrt{v - 1}}{3})^2} \frac{1}{18e^{2u}}\, dv[/tex][Integral] Simplify: [tex]\displaystyle \int {\bigg( 6e^\big{4x} \cdot \frac{e^\big{arctan(3e^{4x})}}{1 + 9e^\big{8x}} \bigg)} \, dx = \frac{3}{2}\int {\frac{\sqrt{v - 1}e^\big{arctan(\sqrt{v - 1})}}{3[1 + v - 1]} \frac{1}{18(\frac{\sqrt{v - 1}}{3})^2}\, dv[/tex][Integral] Simplify: [tex]\displaystyle \int {\bigg( 6e^\big{4x} \cdot \frac{e^\big{arctan(3e^{4x})}}{1 + 9e^\big{8x}} \bigg)} \, dx = \frac{3}{2}\int {\frac{\sqrt{v - 1}e^\big{arctan(\sqrt{v - 1})}}{3v} \frac{1}{2(v - 1)}\, dv[/tex][Integral] Simplify: [tex]\displaystyle \int {\bigg( 6e^\big{4x} \cdot \frac{e^\big{arctan(3e^{4x})}}{1 + 9e^\big{8x}} \bigg)} \, dx = \frac{3}{2}\int {\frac{e^\big{arctan(\sqrt{v - 1})}}{6v\sqrt{v - 1}} \, dv[/tex][Integral] Rewrite [Integration Property - Multiplied Constant]: [tex]\displaystyle \int {\bigg( 6e^\big{4x} \cdot \frac{e^\big{arctan(3e^{4x})}}{1 + 9e^\big{8x}} \bigg)} \, dx = \frac{1}{4}\int {\frac{e^\big{arctan(\sqrt{v - 1})}}{v\sqrt{v - 1}} \, dv[/tex]Step 7: Integrate Pt. 6
Set variables for u-substitution #3.
Set z: [tex]\displaystyle z = arctan(\sqrt{v - 1})[/tex][z] Differentiate [Arctrig Differentiation, Chain Rule]: [tex]\displaystyle dz = \frac{1}{2v\sqrt{v - 1}} \ dv[/tex]See attachment for rest of work (would not fit entire answer in answering box).
Related Questions
The graph of y= 8x² + 7x - 5/5x² + 2x + 6 has a horizontal asymptote with equation (enter the equation of the horizontal asymptote)
Answers
The horizontal asymptote of the rational function y = (8 · x² + 7 · x - 5) / (5 · x² + 2 · x + 6) is y = 8 / 5.
How to determine the equation of a horizontal asymptote
A horizontal asymptote of a rational function is an horizontal line equations to which the rational function tends to when x tends to the infinite. The horizontal asymptote exists whether the lead grade of the numerator is not greater than the lead grade of the denominator.
Now we proceed to find the value of the horizontal function by using limit properties for rational functions:
y = 8 / 5
The horizontal asymptote of the rational function y = (8 · x² + 7 · x - 5) / (5 · x² + 2 · x + 6) is y = 8 / 5.
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Subtract the following polynomials, then place the answer in the proper location on the grid.
2x^2+5x-10
3x2 - x + 9
Answers
Answer:
[tex]-x^2+6x-19[/tex]
Step-by-step explanation:
Step one:
given
[tex]2x^2+5x-10\\\\3x^2 - x + 9[/tex]
we want to perform a subtraction operation
[tex]2x^2+5x-10-(3x^2 - x + 9)\\\\ 2x^2+5x-10-3x^2 + x - 9 \\\\[/tex]
collect like terms
[tex]2x^2-3x^2+5x+x-10-9\\\\-x^2+6x-19[/tex]
49 students are enrolled in health and 21 students are not what percentage of students in the school are in rolled in Health 
Answers
Answer:
im pretty sure its 30%. because 70% are in heath. so 30% of the students arent. i think... hope this helped.
Step-by-step explanation:
Answer:
i'm pretty sure its 30%.
Step-by-step explanation:
because 70% are in heath. so 30% of the students arent. i think... hope this helped.
The mean of four number is 30. if three of the numbers are 12, 50, and 48 what's the fourth number?
Answers
Answer:
if the mean of 4 numbers is 30 then that means that the total of all the numbers added together must add up to 120 (30 x 4)
then you subtract the three numbers that you have from the total
120-12-50-48 = 10
so the last number is 10
Step-by-step explanation:
A store pays by the ounce to have different types of bracelets made and wants categorize them by price drag each purchase to the category that shows the ratio of dollars
Answers
Answer:
include a picture
Answer:
THIS IS THE CORRECT ANSWER FOR IMAGINE MATH BRAINLIEST PLZ
Step-by-step explanation:
yw!
PLEASE HELP? What is the graph form of them equation?
Answers
A hula hoop has a diameter of 44 inches. What is the circumference of the hula hoop? Use 3.14 for tt
Answers
Answer:
i think 126 sorry if im wrong
Step-by-step explanation:
If there are four quarters in a dollar, which expression correctly shows how you might figure out how many quarters are in $9.25
pls help me show my work and the answer :( tysm!!!
Answers
Answer:
37
Step-by-step explanation:
4(9) + 1 = 37
Answer:
There are 37 quarters in $9.25
Step-by-step explanation:
.25 x 37 = 9.25
or
.25 +.25+.25+.25+.25+.25+.25+.25+.25+.25+.25+.25+.25+.25+.25+.25+.25+.25+.25+.25+.25+.25+....till 37
To solve an equation you use the_____________ operation
Answers
Answer:
order of
Step-by-step explanation:
A wall clock is generally a more precise tool for measuring a length of time than a stopwatch.
True
or
False
Answers
Answer:
False
Step-by-step explanation:
Evaluate the numerical expression: 44
The second 4 is an exponent
Answers
its 256.............................
Max purchased a solar pool cover for his swimming pool. What is the area of the
solar cover for the swimming pool?
10ft
32 ft
Square Feet
Answers
Answer:
320 ft^2
Step-by-step explanation:
Please someone help me: Complete the reasons for the proof.
Given:
m ∠3 = m ∠4
To Prove:
∠1, ∠2 are supplementary
Answers
Step-by-step explanation:
1) given
2) 2 non - common sides in 1 ray
3) supplemental
5) opposite angles are congruent
Reasons:
(1) Given
(2) their 2 non-common sides in 1 ray
(3) definition of supplemental angles
(4) Substitution
(5) Opposite angles are congruent (equal)
(6) Substitution
(7) Definition of supplementary angles
Juanita makes $26 an hour at work. She has to take time off work to purchase her skirt, so each hour away from work costs her $26 in lost income. Assume that returning to work takes Juanita the same amount of time as getting to a store and that it takes her 30 minutes to shop. As you answer
the following questions, ignore the cost of gasoline and depreciation of her car when traveling.
Answers
Answer: Local Department Store: $6.5, $118.5
Across Town: $13, $102
Neighboring City: $26, $96
Step-by-step explanation:
Local Department Store: 1 hr=60 min.
60/15=4
26/4=$6.5
112+6.5=$118.5
Across Town: 60/30=2
26/2=$13
89+13=$102
Neighboring City: 60/60=1
26/1=$26
70+26=$96
what is 7 7/8 dvided by 3 1/8 mixed numbers
Answers
Answer:
2.52
Step-by-step explanation:
Answer:
2.52
Step-by-step explanation:
If a student is taking classes, math, English, P.E., history, biology and Spanish, then what is the chance that the student will take math period? Assume that there are six periods in one day.
10%
17%
25%
60%
Answers
Answer:
60%
Step-by-step explanation: i hope this heple
\
WILL GIVE BRAINLIEST!!
Ignore the $
Meyer rolls two fair, ordinary dice with the numbers $1,2,3,4,5,6$ on their sides. What is the probability that at least one of the dice shows a square number?
Answers
Two numbers on each die are squares: 1 and 4. 4 numbers on each die are not squares: 2 , 3, 5, and 6. there are 16 ways for him to roll a non-square die, out of 36 outcomes for the pair of dice. The other 20 options each have a square number showing on one or both dice. So, the probability is 5/9. hope this helps please pick my answer as brainliest, have a good day!
(04.01 MC) A candle burns down at the rate of 0.5 inches per hour. The original height of the candle was 10 inches. Part A: Write a list of 6 ordered pairs to show the height of the candle in inches (y) as a function of time in hours (x) from the first hour after it started burning. For example, the point (0, 10) would represent a height of 10 inches after 0 hours. Explain how you obtained the ordered pairs. (5 points) Part B: Is this relation a function? Justify your answer using the list of ordered pairs you created in Part A. (2 points) Part C: If the rate at which the candle burned was 0.75 inches per hour instead of 0.5 inches per hour, would the relation be a function? Explain your answer using input and output values. (3 points)
Answers
The ordered pairs, justification of the function with different order pairs is required.
The ordered pairs are [tex](0,10),(1,9.5),(2,9),(3,8.5),(4,8),(5,7.5)[/tex]
All values of [tex]x[/tex] has a unique value in [tex]y[/tex] so, it is a function.
y = Height of candle
x = Number of hours
m = Rate = 0.5 inches per hour
c = y intercept = Initial height = 10 inches
The required function is
[tex]y=-0.5x+10[/tex]
Six ordered pairs would be
[tex]x=0,1,2,3,4,5[/tex]
[tex]y=-0.5\times 0+10=10[/tex]
[tex]y=-0.5\times 1+10=9.5[/tex]
[tex]y=-0.5\times 2+10=9[/tex]
[tex]y=-0.5\times 3+10=8.5[/tex]
[tex]y=-0.5\times 4+10=8[/tex]
[tex]y=-0.5\times 5+10=7.5[/tex]
The ordered pairs are
[tex](0,10),(1,9.5),(2,9),(3,8.5),(4,8),(5,7.5)[/tex]
It can be seen that all values of [tex]x[/tex] has a unique value in [tex]y[/tex] so, it is a function.
If m = 0.75 inches per hour
The function would change to
[tex]y=-0.75x+10[/tex]
Some ordered pairs would be
[tex]y=-0.75\times 0+10[/tex]
[tex]y=-0.75\times 1+10[/tex]
[tex](0,10),(1,9.25)[/tex]
The same is seen in this function, all values of [tex]x[/tex] has a unique value in [tex]y[/tex] so, it is a function.
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Solve for x for the triangle
Answers
Answer:
(appx) 4.3
Step-by-step explanation:
[tex] \sin(64) = \frac{x}{10} [/tex]
[tex] = > 0.43 = \frac{x}{10} [/tex]
[tex] = > x = 4.3[/tex]
You have already saved $55. You earn $9 per hour babysitting. You are saving for a camera that costs $199. Write and solve an inequality represents the possible numbers you need to work to buy the camera.
Answers
Write your own situation about the inequality below. 10 x + 20 ≤ 150
Answers
Answer:
k
Step-by-step explanation:
10)×20+21-150
you will end up getting your answers
I want you to just do the easy bit
the calculations if that's alright
What is the image of (-3, -5) after a reflection over the x-axis
Answers
Answer:
(3,-5)
Step-by-step explanation:
I'm pretty sure sorry if I'm wrong
Only the y coordinate changes
help me lol
50 points for whoever answers correctly
Answers
Answer:
C
Step-by-step explanation:
You want to choose the number line graph that appropriately represents the inequality 0 < x ≤ 26.
Graphing inequalitiesThe given compound inequality is conveniently written with the "arrows" pointing to the left. This means the lower limit of the variable is on the left side of the inequality, just as it is on the left end of the number line graph.
The inequality symbols used are "less than" (<) and "less than or equal to" (≤). The "less than" symbol is represented on a number line graph by an open circle. The "less than or equal to" symbol is represented by a filled dot on the number line.
This means you are looking for a graph that has an open circle on the left end of a solid line, and a filled dot on the right end. Only graphs B and C have finite-length lines, and only graph C has a filled dot at the right end.
Car seatsIn English, the word "between" often implies "and including" when it is describing an interval between two limits. Here, you are given 3 car seat specifications:
A: 0 < x ≤ 26 . . . . . up to 26 lbsB: 18 ≤ x ≤ 35 . . . . . between 18 and 35 poundsC: 26 ≤ x ≤ 75 . . . . . between 26 and 76 poundsThe seats appropriate for a 33-lb child would be seats B and C, because 33 is included in each of those intervals.
Select all the expressions which have 3 as the greatest common factor of the terms.
b + 3 + 6c
27n + 66p
38 – 9j + 6k
12x + 75y + 21
–32 – 24z
Answers
Answer:
27n+66p and 12x+75y+21
Step-by-step explanation:
The greatest common factor is 3 for 27n + 66p and 12x + 75y + 21.
What are Factors?Factors of a number or an algebraic expression are the numbers or expressions which divides the given number evenly.
For example, 10 = 2 × 5
So we can say that 2 and 5 are factors of 10.
Greatest common factor of two numbers or expressions, as the name suggests is the greatest of all the common factors of the given numbers or expressions.
Consider 27n + 66p
Factors of 27n are 1, 3, 9 and 27
Factors of 66p are 1, 2, 3, 6, 11, 22, 33 and 66
Greatest common factor of both = 3
Consider 12x + 75y + 21.
Common factors of three of them are 1 and 3.
Greatest common factor is 3.
Hence the expression with greatest common factor as 3 are 27n + 66p and 12x + 75y + 21.
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Find the scale factor of trapezoid QRST if it’s dilated to Q’R’S’T’. Type your result in the empty box provided below. HELPP PLEASE!!!!!!
Answers
Answer:
5 units
Step-by-step explanation:
its hard to tell but i think it dilated 5 units
We want to find the scale factor at which QRST is dilated to form Q'R'S'T'
We will find that the scale factor is 2.
In a general dilation, we will see that all the measures change in the same way, by the scale factor.
This means that:
QR*k = Q'R'
RS*k = R'S'
And so on, where k is the scale factor.
Then we can just find one of the equivalent measures in both trapezoids and find the value of k.
For example, we have:
Q'R' = 8
QR = 4
Replacing these on the above equation we get:
4*k = 8
k = 8/4 = 2
k = 2
The scale factor of dilation is 2.
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Use the percent formula, A = PB, where A is P percent of B, to answer the following question. What is 24% of 60?
Answers
It should be noted that 24 percent of 60 is 14.4.
How to calculate the percentage?The question is to calculate 24% of 60.
This will go thus:
= 24% × 60
= 24/100 × 60
= 0.24 × 60
= 14.4
Therefore, 24% of 60 is 14.4.
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(7-6)•w
simplify the variable expression
Answers
Answer:
7w-6w
Step-by-step explanation:
I think this is what your asking for. If it is, you distribute the w
Answer:
w
Step-by-step explanation:
You complete the subtraction in the parentheses first, then multiple that by w. So:
(7-6) x w
(7-6) = 1
1 x w = w
Angles that form a linear pair
add up to [?]°
Answers
Answer:
180°
Step-by-step explanation:
Two adjacent angles whose sum add up to 180°are called Linear pair. The uncommon arms of two adjacent angles form a straight line.
Use the number line to identify the least value, first quartile, median, third quartile, and greatest value of the data.
Ages of teachers (in years): 30, 62, 26, 35, 45, 22, 49, 32, 28, 50, 42, 35
Answers
Answer: least value - 22
first quartile - 29
median - 35
third quartile - 47
greatest value - 62
Step-by-step explanation:
Burj Khalifa in Dubai is the tallest building in the world, standing at 828 meters. An adjacent
building, 100 meters away, stands at 550 meters tall. What is the angle of depression from Burj
Khalifa to the adjacent building?
Answers
Answer:
Approximately 70 degrees
Step-by-step explanation:
If we make a right triangle connecting the two buildings, we get a triangle with height 278m and length 100m
We can connect these two values to the theta by using the inverse tan formula
arctan(278/100) = 70.21572110836 degrees.
The angle of depression is approximately 70 degrees