Answer:
38
Step-by-step explanation:
3+8=11
11<38
Round to the nearest hundredth 13.711309201
Answer: 13.71
Step-by-step explanation:
Hundredths is the second place
Answer:
13.71
Step-by-step explanation:
The hundredths place is the second decimal, and, since the decimal after that is a 1, you round down and get 13.71
Net A has 3 rectangles with 2 triangles on the side. Net B has 3 rectangles with 2 triangles on the sides.
Students in Mrs. Garcia’s class were practicing drawing nets. She gave them the three-dimensional figure shown. Alina drew the net labeled A and Benjamin drew the net labeled B. What statement about the students’ nets is true?
Alina drew the only correct net.
Benjamin drew the only correct net.
Both nets are correct.
Neither student’s net is correct.
Answer:
c
Step-by-step explanation:
Answer:
~c~
Step-by-step explanation:
Instructions: Name the type of angle relationship. Set up the
equation and solve for x.
X + 89
850
»
Angle Relationship: These are
angles.
Equation:
Solve: 2
Check
Answer:
interior alternative angles
x=--4°
Step-by-step explanation:
please mark me as brainlest
can someone answer real quick
Answer:− -19s
Step-by-step explanation:
Move
I need to know this hard wuestion!
The value '0' is between the points above sea level and points below sea level. '0' being neither negative nor positive means it cannot be below or above sea level, rather '0' meters represents sea level.
Hope that helps!
The differential equation dy dx equals the quotient of the quantity x plus 1 and y minus 3
produces a slope field with vertical tangents at y = 3
produces a slope field with vertical tangents at x = −1
produces a slope field with rows of parallel segments
I only
II only
III only
I and III
Answer:
I only
Step-by-step explanation:
dy/dx = x + 1 / y -3
(y - 3)dy = (x + 1) dx
integrate
1/2y^2 - 3x = 1/2x^2 + x
1/2y^2 = 1/2x^2 + 4x
y^2 = x^2 + 8x
y = (x^2 + 8x)^1/2
Researchers from the Educational Testing Service (ETS) found that providing immediate feedback to students answering openended questions can dramatically improve students’ future performance on exams ( Educational and Psychological Measurement, Feb. 2010). The ETS researchers used questions from the Graduate Record Examination (GRE) in the experiment. After obtaining feedback, students could revise their answers. Consider one of these questions. Initially, 50% of the students answered the question correctly. After providing immediate feedback to students who answered incorrectly, 70% answered correctly. Consider a bank of 100 open-ended questions similar to those on the GRE. (b) After providing immediate feedback, what is the probability that more than 10 of the students answer the question correctly? Group of answer choices 58.8% 41.2% 95.2% 4.8%
Complete question is;
Researchers from the Educational Testing Service (ETS) found that providing immediate feedback to students answering openended questions can dramatically improve students’ future performance on exams (Educational and Psychological Measurement, Feb. 2010). The ETS researchers used questions from the Graduate Record Examination (GRE) in the experiment. After obtaining feedback, students could revise their answers. Consider one of these questions. Initially, 50% of the students answered the question correctly. After providing immediate feedback to students who answered incorrectly, 70% answered correctly. Consider a bank of 100 open-ended questions similar to those on the GRE. a. In a random sample of 20 students, what is the probability that more than half initially answer the question correctly? b. Refer to part a . After providing immediate feedback, what is the probability that more than half of the students answer the question correctly?
Answer:
A) 41.2%
B) 95.2%
Step-by-step explanation:
This is a binomial probability distribution problem, so we will use the formula;
P(k) = (n!/(k!(n - k)!) × p^(k) × (1 - p)^(n-k)
A) Initially, 50% of the students answered the question correctly.
Thus, p = 0.5
Also,n = 20
Now, the probability that more than half initially answer the question correctly would be:
P(k > 10)
This can be expressed as;
P(k > 10) = P(11) + P(12) + P(13) + P(14) + P(15) + P(16) + P(17) + P(18) + P(19) + P(20)
P(11) = (20!/(11!(20 - 11)!) × 0.5^(11) × (1 - 0.5)^(20-11) = 0.1602
Similarly,
P(12) = (20!/(12!(20 - 12)!) × 0.5^(12) × (1 - 0.5)^(20-12) = 0.1201
P(13) = (20!/(13!(20 - 13)!) × 0.5^(13) × (1 - 0.5)^(20-13) = 0.0739
Using online binomial probability calculator we can get the remaining values which are;
P(14) = 0.037
P(15) = 0.0148
P(16) = 0.0046
P(17) = 0.0011
P(18) = 0.0002
P(19) = 0.00002
P(20) = 0
Thus;
P(k > 10) = 0.1602 + 0.1201 + 0.0739 + 0.037 + 0.0148 + 0.0046 + 0.0011 + 0.0002 + 0.00002 + 0 ≈ 0.41192 = 41.2%
B) After providing immediate feedback, we are told that 70% answered correctly.
Thus; p = 70% = 0.7
Similar to A above and using online binomial probability calculator, we have;
P(11) = 0.0654
P(12) = 0.1144
P(13) = 0.1643
P(14) = 0.1916
P(15) = 0.1789
P(16) = 0.1304
P(17) = 0.0716
P(18) = 0.0278
P(19) = 0.0068
P(20) = 0.0008
Thus;
P(k > 10) = 0.0654 + 0.1144 + 0.1643 + 0.1916 + 0.1789 + 0.1304 + 0.0716 + 0.0278 + 0.0068 + 0.0008 = 0.952 = 95.2%
Will Mark Brainliest!!!
A diver is standing on a platform 24 ft above a pool. He jumps from the platform with an initial
upward velocity of 8 ft/s. Using the formula, h(t) = -16t2 + vt + s, where h is his height
above the water, t is the time, v is his starting upward velocity, and s is his starting height.
A) After how long will the diver reach its maximum height?
B) What is the maximum height the diver will reach?
Answer:
he Formula h = -16t^2 + vt + s; where
h = height after t seconds
t = time in seconds
v = upward velocity
s = initial height (t=0)
also note that
-16t^2 represents the downward force of gravity for t seconds
+vt = the upward force for t seconds
:
the height when he hits the water = 0 therefore:
0 = -16t^2 + 8t + 24
usually written
-16t^2 + 8t + 24 = 0
we can simplify this, divide by -8, then we have
2t^2 - t - 3 = 0
this will factor to
(2t - 3)(t + 1) = 0
the positive solution
2t = 3
t = 3/2
t = 1.5 seconds to hit the water
-
Form the actual equation
y=-16t²+vt+sy=-16t²+8t+24Find vertex as vertex is maximum
x coordinate
-b/2a-8/2(-16)8/321/4s0.25sy coordinate
h(0.25)-16(0.25)²+8(0.25)+2425mMax height is 25m
Time to reach is 0.25s
The measure of angle B is 68 degrees. What is the measure of its supplementary angle?
112
suplementry angle add to 180 so 180_68=112
Exercises
1.
Create a real-world situation that relates to the points shown in the number line model. Be sure to describe the
relationship between the values of the two points and how it relates to their order on the number line.
0
A real-world situation to model the number line is: It was colder on Friday night than Saturday morning because 0 is greater than -1
How to interpret the number line?See attachment for the number line
From the figure, we have the following points
0 and -1
The number line can be modelled by temperature where a colder temperature is represented by 0, while the other temperature is -1 degrees Celsius
Hence, a complete scenario is:
On a Friday night, the measure of the temperature is 0 degrees Celsius. On Saturday morning, the temperature is -1 degree Celsius. It was colder on Friday night than Saturday morning because 0 is greater than -1
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EASY POINTS. all irregular/wrong answers will be reported and removed, this is not easy points if you don't know how to factor.
I need these to be in the greatest common factor form, i am in a business calc class and understand everything but the algebra :) i hate algebra.
1) 36x^2-24x
2)12x^2-24x
3)45x^3-30x
Step-by-step explanation:
1)
Take 12x as a common factor
12x(3x-2)
12 since both 36 and 24 are divisible by 12
2)
Take 12x as a common factor again
12x(x-2)
3)
Take 15x as a common factor
15x(3x²-2)
3x²-2 can be written as the difference of 2 squares
15x(√3 x+√2)(√3 x-√2)
Answer:
[tex]12x(3x-2); 12x(x-2); 15x(3x^2-2)[/tex]
Step-by-step explanation:
Look at the coefficients of the first espression. 36 and 24 are both multiples of 12. We can collect 12x from both and we get
[tex]36x^2-24x=12x(3x-2)[/tex]
Second, 12 and 24 are obviously one twice the other, the common factor is again 12x:
[tex]12x^2-24x = 12x(x^2-2)[/tex]
Third polynomial: this time the greatest common divisor of 45 and 30 is 15. Let's collect 15x from either
[tex]45x^3-30x=15x(3x^2-2)[/tex]
What is the value of x in the triangle?
A: 10/2
B: 10
C: 20
D 20/2
E: 10/3
Answer:
[tex]A. 10\sqrt{2}[/tex]
Step-by-step explanation:
The triangle is a 45-45-90 triangle, which has the property that the side across from the 90°, in this case, x, is equal to one of the other sides multiplied by [tex]\sqrt{2}[/tex].
So:
10 x [tex]\sqrt{2}[/tex] = [tex]10\sqrt{2}[/tex]
The value of x in the given triangle is 10√2. So, correct option is A.
In triangle ABC, since angle B is a right angle, we can use the Pythagorean theorem to find the length of side AC (denoted by x).
The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
In this case, AB and BC are the two sides adjacent to the right angle:
AB² + BC² = AC²
Substitute the given values:
10² + 10² = x²
100 + 100 = x²
200 = x²
Now, to find the value of x, take the square root of both sides:
x = √200
Simplify the square root:
x = √(100 * 2)
x = 10√2
So, the value of x in the triangle is 10√2.
The correct answer is A: 10√2.
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Joyce paid $143.00 for an item at the store that was 35 percent off the original price. What was the original price?
Answer:
220
Step-by-step explanation:
GIven:
Joyce paid $143.00 for an item at the store that was 35 percent off the original price.
To Find:
What was the original price?
Solution:
Since that was 35 percent off the original price
Thus, 100 - 35 = 65
Equation:
143 = 0.65x
DIvide both sides by 0.65
143/0.65 = 0.65x/0.65
x = 220
Check Answer:
220 x 65% = 143
~Lenvy~
Long Division: (3x^4-8x^3-x^2+9x+5) / (x-2) pleaseee answer this
3x⁴ = 3x³ • x, and
3x³ (x - 2) = 3x⁴ - 6x³
Subtract this from the numerator to get a remainder of
(3x⁴ - 8x³ - x² + 9x + 5) - (3x⁴ - 6x³) = -2x³ - x² + 9x + 5
-2x³ = -2x² • x, and
-2x² (x - 2) = -2x³ + 4x²
Subtract this from the previous remainder to get a new remainder of
(-2x³ - x² + 9x + 5) - (-2x³ + 4x²) = -5x² + 9x + 5
-5x² = -5x • x, and
-5x (x - 2) = -5x² + 10x
Subtract this from the last remainder to get a new one of
(-5x² + 9x + 5) - (-5x² + 10x) = -x + 5
-x = -1 • x, and
-1 (x - 2) = -x + 2
This gives a new remainder of
(-x + 5) - (-x + 2) = 3
3 does not divide x, so we're done.
So, we have
(3x⁴ - 8x³ - x² + 9x + 5) / (x - 2) = 3x³ - 2x² - 5x - 1 + 3/(x - 2)
Determine whether each of the following statements is true or false. a. The least-squares regression line is the line that makes the square of the correlation in the data as large as possible. False b. The least-squares regression line is the line that makes the sum of the squares of the vertical distances of the data points from the line as small as possible True c. The least-squares regression line is the line that best splits the data in half, with half of the points above the line and half below the line. False d. The least-squares regression line always passes through the point (x-bar,y-bar ), the means of the explanatory and response variables, respectively. True
Answer:
Options A & C are False while Options B & D are true.
Step-by-step explanation:
The definition of least squares regression line is given as;
The Least Squares Regression Line is a line that normally causes the vertical distance from the data points to the regression line to be as small as possible.
The term “least squares” is derived from the fact that the best line of fit is one that minimizes the the sum of squares of the errors which is also called the variance.
From this definition above, we can see that options A and C are false while option B is the perfect definition for the least line of squares.
Now, the point (x-bar,y-bar ) is known as the mean.
However, every least squares must pass through the middle point of the given data. The mean is the middle point of data in regression analysis. Thus, we can infer that the least-squares regression line always passes through the mean which is point (x-bar,y-bar ). Thus, option D is true.
Which expression is equivalent to Measure of angle 4?
Which expression is equivalent to Measure of angle 4?
A triangle has angles 1, 2, 3. The exterior angle to angle 1 is 4, to angle 2 is 5, to angle 3 is 6.
Measure of angle 2 + measure of angle 3
Measure of angle 1 + measure of angle 2
Measure of angle 1 + measure of angle 3
Measure of angle 5 + measure of angle 6
Answer: A
Step-by-step explanation:
Answer:
The answer is A;
✔ m∠2 + m∠3
Maya used a recipe in which 2 tablespoons of vinegar were mixed with 3 tablespoons of olive oil. Which statement about the amount of vinegar mixed with olive oil is true?
2 over 3 tablespoon of vinegar is mixed with every tablespoon of olive oil because the ratio of vinegar to olive oil is 2:3.
2 over 3 tablespoons of vinegar is mixed with every 5 tablespoons of olive oil because the ratio of vinegar to olive oil is 2:5.
2 over 5 tablespoons of vinegar is mixed with every 5 tablespoons of olive oil because the ratio of vinegar to olive oil is 2:5.
2 over 5 tablespoon of vinegar is mixed with every tablespoon of olive oil because the ratio of vinegar to olive oil is 2:3.
Answer:
Step-by-step explanation: becuase if the ratio is 2:3 then the ratio reduced to 1 would be 2/3:1
Answer:
(First Option): 2 over 3 tablespoon of vinegar is mixed with every tablespoon of olive oil
Step-by-step explanation
For every 2 tablespoons of vinegar, it was mixed with 3 tablespoons of olive oil. This leads to a ratio of 2:3, and that rules out option 2 and 3. To get the unit rate we divide 2 tablespoons of vinegar by 3 tablespoons of olive oil. This means that there is 2/3 tablespoons of vinegar for every tablespoon of olive oil and leads it to be answer choice 1.
Critical Thinking: Data Transformation In this problem, we explore the effect on the standard deviation of multiplying each data value in a data set by the same constant. Consider the data set 5, 9, 10, 11, 15.
(a) Use the defining formula, the computation formula, or a calculator to compute s.
(b) Multiply each data value by 5 to obtain the new data set 25, 45, 50, 55, 75. Compute s.
(c) Compare the results of parts (a) and (b). In general, how does the standard deviation change if each data value is multiplied by a constant c?
(d) You recorded the weekly distances you bicycled in miles and computed the standard deviation to be s 3.1 miles. Your friend wants to know the standard deviation in kilometers. Do you need to redo all the calcula- tions? Given 1 mile 1.6 kilometers, what is the standard deviation in kilometers?
Answer:
Kindly check explanation
Step-by-step explanation:
Given the data : 5, 9, 10, 11, 15
s = √[(ΣX - m)² / (n - 1)]
n = number of sample = 5
m = mean
m = ΣX/n
m = (5 + 9 + 10+ 11 + 15) / 5
m = 50/5 = 10
s =√((5-10)^2 + (9-10)^2 + (10-10)^2 + (11-10)^2 + (15-10)^2) / 5-1
= 3.601
(B) Multiply each data value by 5 to obtain the new data set 25, 45, 50, 55, 75. Compute s.
m = ΣX/n
m = (25 + 45 + 50 + 55 + 75) / 5
m = 250/5 = 50
s =√((25-50)^2 + (45-50)^2 + (50-50)^2 + (55-50)^2 + (75-50)^2) / 5-1
= 18. 028
(c) Compare the results of parts (a) and (b). In general, how does the standard deviation change if each data value is multiplied by a constant c?
The standard deviation changes by almost c times multiplied by the initial value.
3.601 * 5 = 18.005 which is almost equivalent to 18.028
(d) You recorded the weekly distances you bicycled in miles and computed the standard deviation to be s 3.1 miles. Your friend wants to know the standard deviation in kilometers. Do you need to redo all the calcula- tions?
No
Given 1 mile 1.6 kilometers, what is the standard deviation in kilometers?
If 1 mile = 1.6km
s = 3.1 miles
s in kilometers = (1.6 * 3.1) = 4.96km
Where is my buddy? Helllo buddy nice to see you
Answer:
[tex]\sf\fbox\red{Answer:-}[/tex]
gud noon!
dopamine is responsible for how we feel pleasure
[tex]\small\fbox{\blue{\underline{mαrk \; mє \; вrαínlíєѕt \; plєαѕє ♥}}}[/tex]
In order for Tiger Woods to get the ball in the hole, he must get it within an area of 14.18 square inches. What is the approximate diameter of a golf hole?
30 points pls show work thank you so much!!!
It can be noted that the diameter will be 4.26 inches.
How to calculate the diameter?From the information given, the area has been given as 14.18 square inches.
Therefore, the diameter will be calculated thus:
πr²= 14.18
3.14 × r² = 14.8
r² = 14.18/3.14
r² = 4.5
r = 2.13
Therefore, the diameter will be;
= 2r
= 2 × 2.13
= 4.26 inches.
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helppppppppppppppppp
Answer:
31-32
Hyposis: SHe is the mother.
Conclusion: She's not.
Step-by-step explanation:
What is the prime factorization of 32?
OA) 232
OB) 23
OD) 28
I
Answer:
2 × 2 x 2 x 2 x 2 or 25
Answer:
2*2*2*2*2, which can be also written as [tex]2^5[/tex].
1) A class of 449 students went on a field trip.
They took 11 velvicles, some cars and some
buses. Find the number of cars and the
number of buses they took if each car holds
3 students and each bus hold 55 students.
Answer: A class of 449 students went on a field trip. They took 11 vehicles, some cars and some buses. Find the number of cars and the number of buses they took if each car holds 3 students and each bus holds 55 students.
Step-by-step explanation: they took 8 buses.
55 x 8 = 440
3 x 3 = 9
440 + 9 = 449
What is the inverse of the function f(x) = 2x + 1?
Answer: X minus 1 over 2
Step-by-step explanation:
Use an inverse function calculator
PLEASE SOLVE IT
DONT STEAL POINYS!!
Answer:
A) 3
Step-by-step explanation:
Method 1)Ques: 30/(4√3 + 3√2) = 4√3 - a√2
Rationalise the denominator.
→ 30/(4√3 + 3√2) × (4√3 - 3√2)/(4√3 - 3√2) = 4√3 - a√2
→ (120√3 - 90√2)/(48 - 18) = 4√3 - a√2
→ (120√3 - 90√2)/30 = 4√3 - a√2
→ 120√3 - 90√2 = 120√3 - 30a√2
→ 90√2 = 30a√2
On solving we get,
→ a = 3
Method 2)Ques: 30/(4√3 + 3√2) = 4√3 - a√2
→ 30 = 4√3(4√3) + 4√3(-a√2) + 3√2(4√3) + 3√2(-a√2)
Solve the brackets,
→ 30 = 48 - 4a√6 + 12√6 - 6a
→ - 18 = - 4a√6 - 12√6 - 6a
Put like terms on one side,
→ - 18 + 6a = - 4√6 (a + 3)
Take the common,
→ 3(a - 3) = - 2√6 (a + 3)
→ 3 (a - 3) = 2√6 (-a - 3)
→ [3(a - 3)]/2√6 = - a - 3
Rationalise the denominator,
→ 2√6(3a - 9)/24 = - a - 3
→ (3a√6 - 9√6)/12 = - a - 3
→ √6(a - 3) = - 4a - 12
Do squaring on both sides,
→ [√6(a - 3)]² = - (4a + 12)²
→ 6(a² + 9 - 6a) = - (16a² + 144 + 96a)
Solve the brackets,
→ 6a² + 54 - 36a = - 16a² - 144 - 96a
→ 22a² - 132a + 198 = 0
→ 11a² - 66a + 99 = 0
→ a² - 6a + 9 = 0
Split the middle term,
→ a² - 3a - 3a + 9 = 0
→ a(a - 3) -3(a - 3) = 0
→ (a - 3)(a - 3) = 0
On comparing we get,
→ a = 3 (key to solve this and similar questions is rationalisation of the denominator)
Hence, the correct option is option A) 3.
A ball is thrown straight up from the top of a building that is 400 ft high with an initial velocity of 64 ft/s. The height of the object can be modeled by the equation s ( t ) = -16 t2 + 64 t + 400.
In two or more complete sentences explain how to determine the time(s) the ball is higher than the building in interval notation.
Check the picture below.
the ball is higher than the building when y = 0, or namely when s(t) = 0, any other time is above it, and hmmm when it's that?
[tex]\stackrel{s(t)}{0}~~ = ~~-16t^2+64t+400\implies 0=-16(t^2-4t-25) \\\\[-0.35em] ~\dotfill\\\\ ~~~~~~~~~~~~\textit{quadratic formula} \\\\ 0=\stackrel{\stackrel{a}{\downarrow }}{1}t^2\stackrel{\stackrel{b}{\downarrow }}{-4}t\stackrel{\stackrel{c}{\downarrow }}{-25} \qquad \qquad t= \cfrac{ - b \pm \sqrt { b^2 -4 a c}}{2 a} \\\\\\ t= \cfrac{ - (-4) \pm \sqrt { (-4)^2 -4(1)(-25)}}{2(1)}\implies t=\cfrac{4\pm\sqrt{116}}{2}[/tex]
[tex]t=\cfrac{4\pm 2\sqrt{29}}{2}\implies t= \begin{cases} 2+\sqrt{29}~~\textit{\Large \checkmark}\\ 2-\sqrt{29} \end{cases}~\hfill \boxed{(0~~,~~2\sqrt{29})}[/tex]
notice, we can't use the 2-√29 because is a negative value and "t" is never negati ve, also notice the interval notation, we have parentheses on both sides.
(a) A square has a perimeter of 68 cm. What is the length of each side?
Answer:
17cm
Step-by-step explanation:
The perimeter of a square is the 4 sides added up, so we can just divide 4 by 68 to find out what one length is.
68/4= 17
We can prove this by adding it again.
17+17+17+17=68
5) Justin went cliff diving. He started out on a cliff that
was 8 feet above the water, and ended up 15 feet
below the surface of the water. How far away from his
starting point was he when he finished?
Number
Answer:
23
Step-by-step explanation:
8+15=23
in all, he moved 8 feet to the water, then another 15 feet below the water. this means that he ended up (8+15) feet from where he started.
Answer:
23
Step-by-step explanation:
[tex] \rm\int_{0}^{ \frac{\pi}{2} } \cot(x) \ln(\sec(x)) \\ [/tex]
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]
How do I get the y-coordinates?
Answer:
Plug the x values into the equation and the answers are the y values