Substitute y = tan(x) and dy = sec²(x) dx to transform the integral to
[tex]\displaystyle \int_0^{\frac\pi2} \cot(x) \ln(\sec(x)) \, dx = \frac12 \int_0^\infty \frac{\ln(1+y^2)}{y(1+y^2)} \, dy[/tex]
and we split the integral at y = 1.
Examining the one over [0, 1], expand into partial fractions :
[tex]\dfrac1{y(1+y^2)} = \dfrac1y - \dfrac y{1+y^2}[/tex]
Then
[tex]\displaystyle \int_0^1 \frac{\ln(1+y^2)}{y(1+y^2)} \, dy = \int_0^1 \frac{\ln(1+y^2)}y \, dy - \int_0^1 \frac{y \ln(1+y^2)}{1 + y^2} \, dy[/tex]
In the integral over [1, ∞), substitute y = 1/z :
[tex]\displaystyle \int_1^\infty \frac{\ln(1+y^2)}{y(1+y^2)} \, dy = \int_0^1 \frac{z \ln\left(1+\frac1{z^2}\right)}{1+z^2} \, dz \\\\ = \int_0^1 \frac{z \ln\left(1+z^2\right)}{1+z^2} \, dz - 2 \int_0^1 \frac{z \ln(z)}{1+z^2} \, dz[/tex]
Some terms cancel and we're left with
[tex]\displaystyle \int_0^\infty \frac{\ln(1+y^2)}{y(1+y^2)} \, dy = \frac12 \int_0^1 \frac{\ln(1+y^2)}y \, dy - \int_0^1 \frac{y \ln(y)}{1+y^2} \, dy[/tex]
Use the series expansions of ln(1 + y) and 1/(1 - y) - both are valid for |y| < 1.
[tex]\displaystyle \int_0^1 \frac{\ln(1+y^2)}y \, dy = - \sum_{n=1}^\infty \frac{(-1)^n}n \int_0^1 y^{2n-1} \, dy = -\frac12 \sum_{n=1}^\infty \frac{(-1)^n}{n^2} = \frac{\pi^2}{24}[/tex]
[tex]\displaystyle \int_0^1 \frac{y \ln(y)}{1+y^2} \, dy = \sum_{n=0}^\infty (-1)^n \int_0^1 y^{2n+1} \ln(y) \, dy = -\frac14 \sum_{n=0}^\infty \frac{(-1)^n}{(n+1)^2} = -\frac{\pi^2}{48}[/tex]
Putting everything together, we have
[tex]\displaystyle \int_0^{\frac\pi2} \cot(x) \ln(\sec(x)) \, dx = \frac12\times\frac{\pi^2}{24} - \left(- \frac{\pi^2}{48}\right) = \boxed{\frac{\pi^2}{24}}[/tex]
The length of a square is 4y + 7. What is the perimeter of the square?
Answer:
Step-by-step explanation:
Perimeter of a square = 4 * Length of the square
Given
Length of a square = 4y + 7
Perimeter of the square = 4(4y+7)
Open the parenthesis
Perimeter of the square = 4(4y)+4(7)
Perimeter of the square = 16y + 28
Hence the perimeter of the square is 16y+28
a cylinder has a height of 16 feet and a diameter of 22 feet. what is its volume?
Question :-
A Cylinder has a Height of 16 Feet and a Diameter of 22 Feet . What is its Volume ?Answer :-
Volume of Cylinder is 6084.5 Feet ³.Explanation :-
As per the provided information in the given question, we have been given that the Height of Cylinder is 16 feet and the Diameter is given as 22 feet . And, we have been asked to calculate the Volume of Cylinder .
For calculating the Volume , first we need to find Radius of Cylinder .
Here, we will find the Radius :-
[tex] \bigstar \: \: \boxed{ \sf{ \: Radius \: = \: \dfrac{Diameter}{2} \: }} [/tex]
[tex] \longmapsto \: \: \sf { Radius \: = \: \dfrac{ \: 22 \: }{2} \: } [/tex]
[tex] \longmapsto \: \: \bf { Radius \: = \: 11 \: Feet } [/tex]
Now, we will find Volume of Cylinder :-
[tex] \bigstar \: \: \boxed{ \sf{ \: Volume \: _{Cylinder} \: = \: \pi r^{2} h \: }} [/tex]
[tex] \longmapsto \: \: \sf {Volume \: _{Cylinder} \: = \: \dfrac{22}{7} \: \times \: ({11})^{2} \: \times \: 16 } \\ [/tex]
[tex] \longmapsto \: \: \sf {Volume \: _{Cylinder} \: = \: \dfrac{22}{7} \: \times \: 121 \: \times \: 16 } \\ [/tex]
[tex] \longmapsto \: \: \sf {Volume \: _{Cylinder} \: = \: \dfrac{22}{7} \: \times \: 1936 } \\ [/tex]
[tex] \longmapsto \: \: \sf {Volume \: _{Cylinder} \: = \: \dfrac{22 \: \times \: 1936}{7}} \\ [/tex]
[tex] \longmapsto \: \: \sf {Volume \: _{Cylinder} \: = \: \dfrac{ \: 42592 \:}{7}} \\ [/tex]
[tex] \longmapsto \: \: \bf {Volume \: _{Cylinder} \: = \: 6084.5 \: Feet^{3}} [/tex]
[tex] \underline {\rule {180pt} {4pt}} [/tex]
Formula Used :-
[tex] \Longrightarrow \: \: \boxed{ \sf{ \: Volume \: _{Cylinder} \: = \: \pi r^{2} h \: }} [/tex]
Where ,
R denotes to the RadiusH denotes to the HeightThe volume of the given cylinder.
Solution:[tex]v = \pi {r}^{2} h[/tex]
[tex]v = \pi \times {11}^{2} \times 16[/tex]
[tex]v = 6082.12338 \: {ft}^{3} [/tex]
[tex]v = 6082.12 \: {ft}^{3} (2 \: dp)[/tex]
Therefore, the volume of the given cylinder is 6082.12 cubic feets.
The sum of 10 times a number in 15 is added to 11 times the same number
Answer:
Now, let x equal the number:
10 times the number is 10x.
Sum of 10 times the number and 15, means sum of 10 times the number plus 15, which is 10x + 15.
Eleven times the same number, means 11 times x, which is 11x.
Sum of 10 times a number and fifteen is added to eleven times the same number is therefore: 10x + 15 + 11x = 21x + 15
Step-by-step explanation:
Please help!!!!!!!!!
I need help pls. The question is in the image attached.
Answer:
A. This is a functional relationship
Step-by-step explanation:
y=8x−5
Use the slope-intercept form to find the slope and y-intercept.
Slope: 8
y-intercept: (0,−5)
Any line can be graphed using two points. Select two x values, and plug them into the equation to find the corresponding y values.
x | Y
0 | -5
5/8 | 0
Graph...
On the graph, you can see each x value corresponds with only one y value.
Which data set would be numerical?
Kinds of pets
Eye color
Age
Favorite food
age
Step-by-step explanation:
it's countable because we can't count eye colour
1
Write the number 28 491 correct to the nearest 1000
(Total for Question 1 is 1 mark
((((Asap will give brainliest)))))
Answer:
8000
Step-by-step explanation:
I'm not sure of it
Multiplying the numerator and denominator of 8/10 by creates the equivalent fraction 40/50
Answer:
5
Step-by-step explanation:
1. Multiply it seperatly, 8 x 5 and 10 x 5
2. Put it back into it's fraction form, 40/50
3. there you go! The answer is "Multiplying the numerator and denominator of 8/10 by 5 creates the equivalent fraction 40/50."
These are just some
equivalent fractions of 40/50:
4/5 = 8/10 = 12/15 = 16/20
20/25 = 24/30 = 28/35 = 32/40
Explain how to find the product of 12 3/8 x 6 7/8. Use complete sentences in your answer.
Answer:
you got to multipply those numbers
Step-by-step explanation:
Applications and Problem Solving
Samantha is 80 cm.tall. Nora's height is 4cm less twice the height of Samantha.
How tall is Nora?
Understand: What is asked? HEb. What
are the given facts?
Plan: Express in algebraic expression. a. Solve and check:
d. Show your computation below. (You can use extra sheet if needed) Answer:
If Samantha is 80cm tall and Nora's height is 4cm less twice the height of Samantha then Nora is 156 cm tall
Algebraic expressionAlgebraic expression is made up of variables and constants, along with algebraic operations such as addition and subtraction.
Samantha is 80 cm tall.
Nora's height is 4cm less twice the height of Samantha. Therefore, Nora's height is represented as follows;
2(80) - 4Nora's height = 160 - 4
Nora's height = 156 cm
learn more on algebraic expression here: https://brainly.com/question/953809
Maria is 19 years old. Her age is 4 more than 3 times some number, x.
Answer: x = 5
Step-by-step explanation:
We can set up an equation to solve for x:
3x + 4 = 19
We can subtract 4 from both sides:
3x = 15
We can divide both sides by 3:
x = 5
2x^4-5x^3+x^2+3x+2 for: x=-5
Answer:
If you look it up it should be there
Step-by-step explanation:
google :\
You can buy an escape room pass that allows you 8 visits for a total of $72 . What is the unit rate for the card in dollars per visit?
Answer:
$9 per visit
Step-by-step explanation:
set up equation
variable x = dollers per visit
8x = 72
goal is to isolate the variable
divide both sides by 8 to do so
8x ÷ 8 = 72 ÷ 8
1x = 9
x = 9
What is the rational number equivalent to 3 point 12 with a bar over 12?
Answer:
3 4/33
Step-by-step explanation:
How many ounces are in 14 pounds
Answer: 224 ounces are in 14 pounds
Step-by-step explanation:
1 pound = 16 ounces
*14 *14
14 pounds = 224 ounces
The sum of the measures of the angles below is 180*
Answer:
x = 45
(2x-10) = 80
x+10 = 55
Step-by-step explanation:
so, make an equation to find x:
x+2x-10+x+10 = 180
collect like terms
4x = 180 (divide both sides by 4)
x = 45
sub into the formulas:
x- 10 = 45+10 = 55
2 × 45 - 10 = 80
add them together to see if it is correct:
45 + 55 + 80 = 180
Gwen is bicycling down a path at a constant speed of 15 miles per hour.
How long does it take her to travel 40 miles?
Answer:
Step-by-step explanation:
156 minutes
2 hours 36 minutes is how long it take Gwen!
-1/3x-4/3y=-2 1/3x-2/3y=4 using elimination
-1/3x-4/3y=2
1/3x-2/3y=4
First realize that the coefficient of x is -1/3 and 1/3 which can be eliminated with addition
Adding the two equations together gets
-6/3y=6
Simplify
-2y=6
y=-3
Substitute back into original equation
1/3x-2/3(-3)=4
1/3x+2=4
1/3x=2
x=6
Answer
x=6, y=-3
Check answer
-1/3(6)-4/3(-3) =?= 2
-2+4=2
2=2
Done!
Please can I have help with these questions
what is the greatest common factor of 30 and 48
Answer:
6
Step-by-step explanation:
Answer:
The answer is 6.
Step-by-step explanation:
5x6=30, and 8x6=48. Hope this helps!
h(x)=-5x+3 , find h(5)
Answer:
-22
Step-by-step explanation:
h(5)=-5•5+3
h(5)=-25+3
h(5)=-22
Answer:
-22
Step-by-step explanation:
Replace x with 5 and multiply tge -5 by 5 wich equals -25 and add 3 which would make -22
(2x + 3)(x – 5) + 7 = ax² + bx + c What is the value of a, b, and c?
Answer:
a = 2 , b = - 7 , c = - 8
Step-by-step explanation:
expand and simplify the left side then compare the coefficients of like terms on both sides.
(2x + 3)(x - 5) + 7 ← expand factors using FOIL
= 2x² - 10x + 3x - 15 + 7 ← collect like terms
= 2x² - 7x - 8
compare to ax² + bx + c , then
a = 2, b = - 7 , c = - 8
Multiply. 1 1/2 x 3 2/3
Answer:
5.5
Step-by-step explanation:
Answer: 5 1/2
Step-by-step explanation:
uhh just multiply using the exact value
Can someone help me find the volume of this shape?
Answer:
18
Step-by-step explanation:
To find volume you have to use the formula (length x width x height.)
So by looking at it you can see that it's 3 units long, 3 units in wide, and 2 units tall. Knowing that, you then plug in those numbers to the formula. That would be 3x3x2. Any calculator will work.
#14. Using the information from question #14, if you only want to put 50% of your discretionary monies toward the emergency fund each month, how many months will it take to fully fund it, excluding interest accrual? (Note: Round up to the next whole month.)
#15 Using the information from question #14, if you only want to put 50% of your discretionary monies toward the emergency fund each month, how many months will it take to fully fund it, excluding interest accrual? (Note: Round up to the next whole month.)
To save 6 months’ worth of your fixed expenses in an emergency fund will take approximately 5 months.
What is an emergency fund?An emergency fund is a safety net used specifically to cover basic living expenses for at least 6 months in case future mishaps or unexpected expenses arise.
Financial experts generally recommend six months of basic living (fixed) expenses to put away in an emergency fund.
Data and Calculations:The amount to save in the emergency fund = $3,753.79 ($2,085.44 x 30% x 6)
50% of discretionary monies = $729.90 ($2,085.44 x 1 - 30%) x 50%
Months to save $3,753.79 saving $729.90 = 5 months ($3,753.79/$729.90)
Question Completion:Your realized income is $2,085.44, and your fixed expenses are 30%. You want to save 6 months’ worth in an emergency fund.
How many months will it take to fully fund it, excluding interest accrual?
Thus, o save 6 months’ worth of your fixed expenses in an emergency fund will take approximately 5 months.
Learn more about discretionary expenses and emergency funds at https://brainly.com/question/422550
Answer:
To calculate this, we first need to find the amount of discretionary money left after paying fixed expenses:
Discretionary money = Realized income - (Fixed expenses as a percentage of realized income)
Discretionary money = $2,085.44 - (30% x $2,085.44)
Discretionary money = $2,085.44 - $625.63
Discretionary money = $1,459.81
We want to put 50% of this toward the emergency fund each month, so:
Emergency fund contribution per month = 50% x $1,459.81
Emergency fund contribution per month = $729.91
To find out how many months it will take to fully fund the emergency fund, we divide the total amount needed by the monthly contribution:
Number of months = (Total amount needed) / (Emergency fund contribution per month)
Number of months = $18,292.65 / $729.91
Number of months = 25.06
Rounded up to the nearest whole month, it will take 26 months to fully fund the emergency fund.
Therefore, the answer is:
26 months
Hi is this C, D or other? Which of these pairs are like terms?
20 x, 0.8 x
17, 3 x
14 x, 14 x2
0.5 x, 0.5 y
[tex]\qquad \qquad\huge \underline{\boxed{\sf Answer}}[/tex]
The pair having like terms is ' A '
It has both terms of x, I.e 20x and 0.8x
I’ll mark brainlest if you help with this question please!
Answer:
[tex]solve for x\\\\x=\frac{5}{11}[/tex]
[tex]factor\\(x-5)x(x-7)[/tex]
Step-by-step explanation:
[tex]solve for x\\15=\frac{5}{x}+4\\ -\frac{5}{x}=4-15\\-\frac{5}{x} =-11\\\frac{11x}{11} =\frac{5}{11} \\x=\frac{5}{11}[/tex]
[tex]factor\\x^{2} -12x+35\\x^{2} -5x-7x+35\\x(x-5)-7(5-x) \\ (x-5)*(x-7)[/tex]
Review the table of values for function f(x).
Which number is the value of f1(-2)?
Answer:
D) 1
Step-by-step explanation:
took on edge
Answer:
D-1
Step-by-step explanation:
Ed2021
I WILL GIVE BRAINLISTTTT
Answer: C
Step-by-step explanation:
What’s is the fraction in simplest form ? Of 9/12
Answer:
3/4
Step-by-step explanation:
The smallest number that makes 9 is three the smallest number that makes 12 is 4 because it cant go below 3
Answer:
3/4
Step-by-step explanation:
Both numbers can be divided by three.
9 divided by 3 is 3 and since the 9 was the numerator the three becomes your new numerator.
12 divided by 3 is 4 and since 12 was the denominator the four becomes your new denominator.
Hope this helps! Good Luck!