I assume the natural logarithm, and not the base-10 log. Parameterize the integral as
[tex]\displaystyle I(a) = \int_0^1 \frac{(x^a - 1)^2}{\log^2(x)} \, dx[/tex]
Differentiate twice with respect to [tex]a[/tex].
[tex]\displaystyle I'(a) = 2 \int_0^1 \frac{x^{2a} -x^a}{\log(x)} \, dx[/tex]
[tex]\displaystyle I''(a) = 2 \int_0^1 (x^{2a} - x^a) \, dx[/tex]
Evaluate the last integral, then solve for [tex]I(a)[/tex] with the fundamental theorem of calculus, noting that [tex]I(0)=I'(0)=0[/tex].
[tex]\displaystyle I''(a) = 2\left(\frac1{2a+1} - \frac1{a+1}\right)[/tex]
[tex]\displaystyle I'(a) = 2 \int_0^a \left(\frac1{2t+1} - \frac1{t+1}\right) \, dt \\\\ ~~~~ = \log(2a+1) - 2 \log(a+1)[/tex]
[tex]\displaystyle I(a) = \int_0^a (\log(2t+1) - 2\log(t+1)) \, dt \\\\ ~~~~ = (2a+1) \log(2a+1) - 2(a+1) \log(a + 1)[/tex]
Let [tex]a=\phi[/tex] to recover the original integral.
[tex]\displaystyle I(a) = (2\phi+1) \log(2\phi+1) - 2(\phi+1) \log(\phi + 1)[/tex]
Using various properties of the golden ratio [tex]\phi[/tex], namely
[tex]\phi = 1 + \dfrac1\phi \implies \phi^2 = \phi + 1 \implies \phi^3 = \phi^2 + \phi[/tex]
[tex]2\phi+1 = \phi\left(2+\dfrac1\phi\right) = \phi(1 + \phi)[/tex]
[tex]\phi=\dfrac{1+\sqrt5}2 \implies \phi^3 = 2 + \sqrt5[/tex]
we can simplify the result to
[tex]\displaystyle I(\phi) = \int_0^1 \frac{(x^\phi-1)^2}{\log^2(x)} \, dx = \boxed{\sqrt5\,\log(\phi)}[/tex]
What are some similarities between sales tax and hotel tax? Select all that apply. Multiple select question. cross out A) Both are an additional amount one must pay. cross out B) Both must be paid in advance when you make a hotel reservation. cross out C) Both are based on a percentage of the original cost. cross out D) Both the sales tax and the hotel tax are equal when you stay in a hotel. cross out E) Both represent a guest fee that hotels charge their customers.
Some of the similarities between sales tax and hotel tax include the following:
B) Both must be paid in advance when you make a hotel reservation.
C) Both are based on a percentage of the original cost.
What is taxation?Taxation can be defined as the involuntary fees that are levied on individuals or business firms by the government of a particular country, so as to generate revenues that can be used to fund public projects, institutions and activities.
The types of tax.In Economics, there are different types of tax that are levied on individuals or business firms by the government and these include the following:
Hotel taxSales taxProperty taxIncome taxIn conclusion, we can reasonably infer and logically deduce that both sales tax and hotel tax must be paid in advance when making a hotel reservation in the United States of America.
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Find the median 6, 13, 15, 9, 6, 11, 14
The median of a set of numbers is the number that appears in the middle after arranging the set of numbers in ascending or descending order
Step 1:Arrange the given values in ascending order.
Step 2. Find the number of observations in the given set of data. It is denoted by n.
Step 3. If n is odd, the median equals the [(n+1)/2]th observation.
The set of numbers given in the question are
[tex]6,13,15,9,6,11,14[/tex]By rearranging the numbers in ascending order, we will have
[tex]6,6,9,11,13,14,15[/tex]The numbers are 7 in number that is
[tex]N=7[/tex]Since the value of N is an odd number, then the median will be calculated using the formula below
[tex]\begin{gathered} \text{Median}=\frac{(n+1)}{2}th \\ \text{Median}=\frac{7+1}{2}th\text{ number} \\ \text{Median}=\frac{8}{2}th\text{ number} \\ \operatorname{median}=4th\text{ number} \end{gathered}[/tex]The 4th number after arranging the set of numbers is
[tex]=11[/tex]Hence,
The median of the set of numbers in the question = 11
Find (f◦g)(x) and (g◦f)(x) for the pair of functions and give the domains. f(x)= 16 x2,
Need more information to solve, but the standard rule is:
(f◦g)(x) = f(g(x))
(g◦f)(x) = g(f(x))
Factor-2x² + 10x.
O-2x(x + 5)
O 2x(-x + 5)
O2(-x² + 10x)
Ox(2x + 10)
Answer:
I believe it is B.
Step-by-step explanation:
When you solve B, you get the expression that you are supposed to factor.
7 Describe the error in solving the equation
(1) 3 (X + 4) = 13
(2) 3x + 4 =13
(3) 3 x = 9
(14)X = 3
Answer:
(2) The 3 was not distributed correctly.
3(x + 4) is the same as 3(x) + 3(4) Which is 3x + 12 not 3x + 4
Step-by-step explanation:
Answer: The error was at the first step since 4 was never multiplied by 3, therefore the final answer would be incorrect.
Step-by-step explanation: 3(x + 4) = 13
multiply 3 by everything in the parenthesis, so
3x + 12 = 13
subtract 12 from each side.
3x = 1
divide both sides by 3
x = 1/3 = 0.33
slove using elimination10x + 8 y = 20 10 x + 2 y = -10
we have
10x + 8 y = 20 ------> equation A
10 x + 2 y = -10 ------> equation B
Multiply equation A by -1 both sides
so
-10x-8y=-20 -----> new equation A
Adds new equation A and equation B
\
--------------------
-8y+2y=-20-10
-6y=-30
y=5
Find the value of x
substitute the value of y in equation A or equation B or new equation A
10x + 8(5) = 20
10x+40=20
10x=20-40
10x=-20
x=-2
therefore
the solution is the ordered pair (-2,5)3+2x1/10+4x1/1000 in standard form
Answer:
3.204
Step-by-step explanation:
3 + 0.2 + 0.004
One Serving of pancakes takes 2/3 cups of milk how many cups of milk with 24 servings for pancakes required
Answer:
Step-by-step explanation:
Given that:-
Since, Serving of 1 pancakes takes 2/3 cups of milk
Hence, serving of 24 pancakes takes (2/3)*24
=48/3
=16
So that; 16 cups of milk are required for serving of 24 pancakes.
Answer:
16 cups of milk
Step-by-step explanation:
Make a proportion
[tex]\frac{serrvings}{cups of milk}[/tex] = [tex]\frac{servings }{cups of milk}[/tex]
[tex]\frac{1}{\frac{2}{3} }[/tex] = [tex]\frac{24}{x}[/tex] cross multiply and solve
1x = [tex]\frac{2}{3}[/tex][tex](\frac{24}{1})[/tex] 1x is the same a x. 24 is the same as [tex]\frac{24}{1}[/tex]
x = [tex]\frac{48}{3}[/tex]
x = 16
in a sale normal prices are reduced by 10% Nathaniel bought a pair of shoes in the sale for £54
what was the original price of the shoes?
Answer:
$59.40
Step-by-step explanation:
54*0.10 (10%) = 5.4
5.4+54= 59.4
therefore, the price of the shoes without the discount is $59.40
Answer:
[tex]x=540[/tex]
What are Percentages?Percents are a ratio of a number and 100. So they are easier to compare than fractions, as they always have the same denominator, 100.
[tex]{\text}To find this.[/tex]
[tex]{\text}Right the formula:[/tex]
10% of x = £54
[tex]{\text}Indicating 10 percent of something is the sale given for the pair of shoes 54.[/tex]
[tex]{\text}Then calculate:[/tex]
[tex]\frac{10x}{100} =54[/tex]
[tex]{\text}Using crisscross multiplication:[/tex]
[tex]x=540[/tex]
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Hello I need help please
To graph the function using a table of values. The function is:
[tex]f(x)=\sqrt[]{x}[/tex]Since the root is even, because it is a square root, there is no Real values for
[tex]x<0[/tex]This already indicates that the probable correct alternative is C, however let's make the table to make sure.
The best way is to get x values that will easily get was f(x) values. Let's pick x = 0, x = 1 and x = 4:
[tex]\begin{gathered} f(0)=\sqrt[]{0}=0 \\ f(1)=\sqrt[]{1}=1 \\ f(4)=\sqrt[]{4}=2 \end{gathered}[/tex]So, the points (0,0), (1,1) and (4,2) should be in the line. This exclude alternatives A, B and D. In C we can see these points:
So, the graph of the given function is the one showed in option C
You need 600 mL of a 20% alcohol solution. On hand, you have a 60% alcohol mixture. How much of the 60% alcohol mixture and pure water will you need to obtain the desired solution?
Answer: there 3,000 total milliliters accourding to my calculation
there for your looking for 60% so your answer should be 1800 since 60% of 3,000 is 1800
1800 ML = 60%
Step-by-step explanation:
the math is above
What system of linear inequalities is shown in the graph?
The system of inequalities is:
y < -4*x + 1
y ≥ (1/2)*x + 1
Which system of linear equations is graphed here?Here we can see two lines, one dashed that passes through (0, 1) and (-1, 5), then the slope is:
s = (5 - 1)/(-1 - 0) = -4
And the y-intercept is 1, then the inequality is:
y < -4*x + 1
The other line is solid, and the shaded region is above the line.
This line passes through (0, -1) and (2, 0), then the slope is:
s = (0 + 1)/(2 - 0) = 1/2
So the inequality is:
y ≥ (1/2)*x + 1
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are y=5x-4 and y=-5x+5 parallel or perpendicular
Answer:
Perpendicular
Step-by-step explanation:
It js seems right. i did one very similar and the awnser was Perpendicular
At a speed of 7.2 km per 2 hours how far can you drive for 7 hours
A. 357
B.358.50
C.359
Answer:
Step-by-step explanation:
divide 7.2 by 2 so you have an hourly rate. Then you have 3.6, multiply that by 7 to get 25.2. Assuming its not in km cause that wouldn't make sense you would just have to convert it into whichever unit it would take
2(4x-3)-8-2x=4
someone please help me with this, fast
....................... .x=3
need asap TIA bonus brownie points if you edit the graph <3
From the following figures, we have that:
a) The sequence of transformations that maps triangle ABC to triangle A''B''C'' is given by:
Reflection over the x-axis.Translation of 11 units right and 1 unit up.b) The rules are given as follows:
Reflection: (x,y) -> (x, -y).Translation: (x,y) -> (x + 11, y + 1).What are transformations on the graph of a function?The transformations used in this problem are as follows:
Translation: Can be in the direction left/right or down/up, changing just the position of the figure.Reflections: Can be a reflection over one of the axes or over a line, changing the orientation of the figure.For this problem, the orientation of the triangle changed, with the C at the bottom of the triangle being transformed into C'' at the top of the transformed triangle, hence the figure was reflected over the x-axis, and the rule is:
(x,y) -> (x, -y).
After the reflection, the coordinates of C' are given as follows:
C' (-4, -1).
The final coordinates of C'' are given as follows:
C'' (7, 0).
Meaning that it was translated 11 units right and 1 unit up, and the rule is given by:
(x,y) -> (x + 11, y + 1).
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what is the slope of the line?
Answer:
slope :
[tex] \dfrac{y2 -y1}{x2 - x1} [/tex][tex] \dfrac{3 - 1}{0 - 1} [/tex][tex] \dfrac{2}{ - 1} [/tex][tex] - 2[/tex]Correct option - F
The ages of the signers of the Declaration of Independence are shown. (Age is approximate since only the birth year appeared in the source, and one has been omitted since his birth year is unknown.) Construct a grouped frequency distribution and a cumulative frequency distribution for the data, using 7 classes.
Answer:
Step-by-step explanation:
The ages of the signers of the Declaration of Independence are shown. (Age is approximate since only the birth year appeared in the source, and one has been omitted since his birth year is unknown.) Construct a grouped frequency distribution and a cumulative frequency distribution for the data using 7 classes. Limits Boundaries 27-33 26.5-33.5 7 41 54 47 40 39 35 50 37 49 42 70 32 44 52 39 50 40 30 34 69 39 45 33 42 44 63 60 27 42 34 50 42 52 38 36 45 35 43 48 46 31 27 55 63 46 33 60 62 35 46 45 34 53 50 50 34-40 33.5-40.5 14 41-47 40.5-47.5 15 48-54 11 47.5-54.5 54.5-61.5 55-61 3 62-68 61.5-68.5 3 69-75 68.5-75.5 55
Please answer question 5 and 6.
5.Given the functions f(x)=-2x + 4 and g(x) = 2x- 8, find the value of x for which f(x) = g(x).
6.Graph the function.
By graphing both equations and finding where the graphs intercept, we can see that the solution for the system is x = 3.
For which value of x both functions give the same output?Here we have the functions:
f(x) = -2x +4
g(x) = 2x - 8
We want to find the value of x such that:
f(x) = g(x).
If we graph the two functions, we just need to identify the x-value of the point where the functions intercept.
The graph can be seen in the image below.
There we can see that the graphs intercept at (3, -2), meaning that the solution of:
f(x) = g(x) is x = 3.
Which is the same thing that if we write:
f(x) = g(x)
-2x + 4 = 2x - 8
4x = 12
x = 12/4 = 3
x = 3
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The numerator of a rational number is 4 less than its denominator. If I is added to both its numerator and denominator, it becomes 2/3 Find the rational number.
The required rational number is 7/11.
let the denominator be x.
and since we are given that the numerator of a rational number is 4 less than its denominator, the numerator will be x-4.
working on the given condition, i.e. If I is added to both its numerator and denominator, it becomes 2/3
(x-4+1)/(x+1) = 2/3
further solving it by cross multiplication we get,
3(x-4+1)=2(x+1)
=> 3(x-3)=2x+2
=> 3x-9=2x+2
=> 3x-2x=9+2
=> x=11
=> x= 11 (denominator)
we know the numerator is 4 less than the denominator i.e.
11-4=7
thus the required fraction will be 7/11.
What are rational numbers?
A rational number is-" one that can be stated mathematically as the ratio or fraction p/q of two integers, p and q, which must both be non-zero".
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Help giving 100 points!!
Given the coordinates of two points on the line, use the slope formula to determine the slope of the line. The slope formula is the quotient of the difference in y-values and the difference in x-values.
a ferry charges $40 for each car and $8.50 for each person in the car. Does this situation represent a function?
Yes, This situation represent a function.
What is linear expression?
A linear expression is an algebraic statement where each term is either a constant or a variable raised to the first power.
Given that;
The cost of each car = $40
The cost of each person = $8.50
Let the number of car = x
The number of persons in a car = z
Then, Total cost of car = $40x
Total cost of persons in car = $8.50z
So, Total cost (y) = $40x + $8.50z
Thus, The function for this situation is;
y = $40x $8.50z
Clearly, This is a linear function with three variable.
Thus, This situation represent a function.
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Missed the class now lost
substitute the given values in the formula of (BAC) we have
[tex]48\cdot3.2\colon160-3\cdot0.015[/tex]Remember
Applying PEMDAS
P ----> Parentheses first
E -----> Exponents (Powers and Square Roots, etc.)
MD ----> Multiplication and Division (left-to-right)
AS ----> Addition and Subtraction (left-to-right)
so
step 1
solve the multiplication
48*3.2=153.6
substitute
[tex]153.6\colon160-3\cdot0.015[/tex]step 2
Solve the division
153.6/160=0.96
substitute
[tex]0.96-3\cdot0.015[/tex]step 3
solve the multiplication
3*0.015=0.045
substitute
[tex]0.96-0.045[/tex]step 4
solve the subtraction
0.96-0.045=0.915
Please answer ASAP for 100 points and brainliest if possible and if its correct
PLEASE!!!!!!!!!!!!!!!!
Answer:
59.40
Step-by-step explanation:
Peter enrolled $80,000 worth of debt with National Debt Relief. Peter’s fee for the service is 20% of the enrolled debt. How much is Peter’s fee?*
Answer:
16000
Step-by-step explanation:
GIVING BRAINLIEST + 60 POINTS!
Solve for x: 1 < x + 3 < 4
A. 4 > x > 7
B. 4 < x < 7
C. −2 > x > 1
D. −2 < x < 1
Please explain why you have gotten the right answer!
Answer:
it's greater than one so it's value is not negative
and 3+4 is 7
so x is greater then 4 but less than 7
so it's b the correct option
Answer:
D. -2 < x < 1
Step-by-step explanation:
1 < x + 3 < 4
=
{1 < x + 3
{x + 3 < 4
=
{ 1 + ( -3 ) < ( x + 3 ) + ( -3 )
{ ( x + 3 ) + ( -3 ) < 4 + ( -3 )
=
{ 1 - 3 < x + 3 - 3
{ x + 3 - 3 < 4 - 3
=
{ -2 < x
{ x < 1
=
{ x > - 2
{ x < 1
Hence, the answer would be : -2 < x < 1
Hopes this helps!
Have a good day :)
{(-3,-4), (-1,2), (0, 0), (-3,5), (2,4)}
Determine whether the relation represents a function.
O Yes, It is a function.
No, it is NOT a function.
Answer:
No, it's not function
Step-by-step explanation:
Every function is a relation but every relation is not a function because:
Function: a special type of relation in which the first-order pairs are different.
In this case {(-3,-4), ......., (-3,5)} these two sets have the same first-order pair.
So, your answer will be 'It's not function.'.
Divide: (u² - 3u - 5) ÷ (u+3).
Answer:
u - 6 + 13/u+3
Step-by-step explanation:
Have a good day :)
4) A done relevant a bomb witha velocity of 20m/s horizotally from a hieght of 50m the ground clculate time
The time taken for the bomb to fall to the ground is 2.5 sec.
What is a projectile motion?In this case, the bomb is moving like a projectile and has projectile motion.A body can only move in a projectile motion when it is affected by gravity. A body moving as a projectile is a projectile motion.We model a particle as having zero acceleration (and thus constant velocity) when it is projected horizontally.The particle in this instance has a zero initial velocity and a constant downward vertical acceleration of g.Given:
Height (h) = 50 m
Horizontal velocity(u) = 20 m/s
Time (t) =?
We use, h = ut to determine the time
50 = 20 × t
Divide both sides by 20,
t = 50 / 20
t = 5/2 s ≅ 2.5 sec
Thus, the bomb will take 2.5 sec to land on the ground.
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The complete question is: "A bomb is projected horizontally downward from a height of 50m with a velocity of 20m/s. Calculate the time taken for the bomb to fall to the ground."
The water pressure on a submerged object is given by P = 64d, where P is the pressure in pounds per square foot, and d is the depth of the object in feet.
a. solve the formula for d.
b. Find the depth of a submerged object if the pressure is 672 pounds per square foot.
a. Solving for 'd' as the subject of formula gives d = P/64
b. The depth of a submerged object if the pressure is 672 pounds per square foot is 10. 5 feet
What is a function?A function can be defined as an expression, law or rule that shows the relationship between two variables.
These variables are namely;
Independent variableDependent variableGiven the function as;
P = 64d
Where;
P is the pressure in pounds per square foot d is the depth of the object in feetNow, let's make the variable 'd' the subject of formula
Divide both sides of the equation with the coefficient of the variable 'd' which is 64. This is done so that the variable stands alone.
We have;
P/64 = 64d/64
d = P/ 64
If the pressure, P equals 672 pounds per square foot
Substitute the value of the variable into the formula, we have;
d = P/64
Then,
d = 672/64
Find the quotient
d = 10. 5 feet.
Thus, the value is 10. 5 feet
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