Answers
Answer:
0.90
Step-by-step explanation:
You want the probability of either event A or event B given the Venn diagram shown.
ProbabilityThe probability of an event shown in the diagram is the ratio of counts for that event to the total of all counts in the diagram. That total is 50.
The number of counts that are not part of either event is shown as 5.
P(A or B) = 1 - P(not (A or B)) = 1 - 5/50
P(A or B) = 0.90
__
Additional comment
Using the given formula, ...
P(A or B) = (25+5)/50 +(15 +5)/50 -5/50 = (25 +15 +5)/50 = 45/50
P(A or B) = 0.90
<95141404393>
Related Questions
Out of a pool of 26 men and 29 women what is the probability of 12 women jurors being chosen?
Answers
The probability of 12 women jurors being chosen is approximately 0.000342 or 0.0342%.
To calculate the probability of selecting 12 women jurors out of a pool of 26 men and 29 women, we need to consider the total number of possible combinations and the specific combination of interest.
The total number of ways to choose 12 individuals from a pool of 55 (26 men + 29 women) is given by the combination formula:
C(55, 12) = 55! / (12! * (55-12)!) = 22579284062370.
The number of ways to choose 12 women from a pool of 29 is given by the combination formula:
C(29, 12) = 29! / (12! * (29-12)!) = 7726160.
Therefore, the probability of selecting 12 women jurors is the ratio of the number of ways to choose 12 women to the total number of possible combinations:
P(12 women jurors) = C(29, 12) / C(55, 12) ≈ 7726160 / 22579284062370 ≈ 0.000342.
Hence, the probability of 12 women jurors being chosen is approximately 0.000342 or 0.0342%.
For more such questions on women jurors
https://brainly.com/question/24130276
#SPJ8
a bread making machine pours out 50 kg of mixture in 8 minutes in 8 minutes.at what rate is the mixture being poured?
Answers
To calculate the rate at which the mixture is being poured, we divide the amount of mixture poured (in kilograms) by the time it takes (in minutes):
Rate = Amount of mixture / TimeIn this case, the amount of mixture poured is 50 kg, and the time taken is 8 minutes. Plugging these values into the formula, we have:
Rate = 50 kg / 8 minDividing 50 kg by 8 min gives us:
Rate = 6.25 kg/minTherefore, the mixture is being poured at a rate of 6.25 kilograms per minute.
TOURISM At a town with a large convention center, the cost of a hotel room has increased 5.1% annually.
If the average hotel room cost $48.00 in 1980 and this growth continues, what will an average hotel room
cost in 2012? Use y = a(1 + r)^t and round to the nearest cent.
a. $143.38
h
$235.79
$126 34
d
$87 19
Answers
Given statement solution is :- The average hotel room cost in 2012 would be $199.97, none of the provided options are correct.
To calculate the average hotel room cost in 2012, we can use the formula:
y = a(1 + r)^t
where:
y = future value (cost in 2012)
a = initial value (cost in 1980)
r = annual growth rate (5.1% or 0.051)
t = number of years (2012 - 1980 = 32)
Substituting the given values into the formula, we have:
y = $48.00(1 + 0.051)^32
Calculating the value:
y ≈ $48.00(1.051)^32
y ≈ $48.00(2.718282)^0.051(32)
y ≈ $48.00(2.718282)^1.632
y ≈ $48.00 * 4.171331
y ≈ $199.97
Rounding to the nearest cent, the average hotel room cost in 2012 would be $199.97.
Therefore, none of the provided options are correct.
For such more questions on Hotel Room Cost in 2012
https://brainly.com/question/14944248
#SPJ8
Out of a pool of 26 men and 29 women what is the probability that 6 men and 6 women will be selected?
Answers
Let's evaluate the probability based on the given information.
The number of ways to choose 6 men from 26 can be calculated as:
C(26, 6) = 26! / (6! * (26-6)!) = 26! / (6! * 20!) = 230,230.
The number of ways to choose 6 women from 29 can be calculated as:
C(29, 6) = 29! / (6! * (29-6)!) = 29! / (6! * 23!) = 177,100.
The total number of possible outcomes is the combination of selecting 6 men and 6 women from the given pool of 26 men and 29 women:
Total outcomes = C(26, 6) * C(29, 6) = 230,230 * 177,100 = 40,795,315,300.
Therefore, the probability of selecting exactly 6 men and 6 women from the given pool is:
Probability = 1 / Total outcomes = 1 / 40,795,315,300 ≈ 2.45 x 10^-11.
Hence, the probability is approximately 2.45 x 10^-11.
Use the binomial theorem to find the expansion of ( x - y + z)^3
Answers
(x - y + z)^3 = C(3,0)x^3 + C(3,1)x^2(-y) + C(3,2)x(-y)^2 + C(3,3)(-y)^3 + C(3,0)x^2z + C(3,1)x(-y)z + C(3,2)(-y)^2z + C(3,0)xz^2 + C(3,1)(-y)z^2 + C(3,0)z^3
where C(n,k) is the binomial coefficient, given by:
C(n,k) = n! / (k! * (n-k)!)
Substituting the values, we get:
(x - y + z)^3 = 1x^3 - 3x^2y + 3xy^2 - 1y^3 + 3x^2z - 6xyz + 3y^2z + 3xz^2 - 3yz^2 + 1z^3
Therefore, the expansion of (x - y + z)^3 using the binomial theorem is:
x^3 - 3x^2y + 3xy^2 - y^3 + 3x^2z - 6xyz + 3y^2z + 3xz^2 - 3yz^2 + z^3.
Solve for x:
A: 3x - 7 = -19; x =
B: 2x + 3 = -11; x =
C: -3 + 1 = -26; x =
D: 6x + 3 = -15; x =
E: -9 + 7x = 33; x =
F: -2x + 8 = 4; x =
( URGENT PLEASE SOLVE !!! )
Answers
A: x = -4
B: x = -7
C: There is no solution for x since -2 does not equal -26.
D: x = -3
E: x = 6
F: x = 2
14. Simon earned 400sh as a for goods sold what Commission 15,000 worth so, 600 could be his varnings for a total sale of 7,000
Answers
His total earnings for a total sale of 7,000 could be 1450sh
How to determine his earnings for a total sale of 7,000.From the question, we have the following parameters that can be used in our computation:
Salary = 400
Commission = 15%
using the above as a guide, we have the following:
Total earnings = 400 + 15% * Total sales
So, we have
Total earnings = 400 + 15% * 7000
Evaluate
Total earnings = 1450
Hence, the total earnings is 1450
Read more about commission at
https://brainly.com/question/26283663
#SPJ1
Question
Simon earned 400sh weekly for goods sold and a commission of 15%
What could be his earnings for a total sale of 7,000.
Graph the relation shown in the table. Is the relation a function? Why or why not?
x:y
-1:3
-1:2
0:-4
4:2
No; a horizontal line passes through two graphed points.
Yes; no vertical line passes through two graphed points.
No; a vertical line passes through two graphed points.
Yes; no horizontal line passes through two graphed points.
Answers
The graph of this relation is four points on the coordinate plane (see attached diagram).
Hence option D is true.
Since we know that,
A relation between a collection of inputs and outputs is known as a function. A function is, to put it simply, a relationship between inputs in which each input is connected to precisely one output.
And a function is a special relation,
where each input x has a single output y.
This relation is not a function, because for one x (x=-1),
Here are two different possible y (y=3 or y=2).
If you consider the graph, you can see that there will be vertical line passing through the points (-1,3) and (-1,2).
To learn more about function visit:
https://brainly.com/question/8892191
#SPJ1
y = x² + 6x - 12
a =______
b = _____
c=______
h=______
k =_____
y-intercept=______
x-intercepts =______
and_____
Answers
Answer:
See below for answers and explanations
Step-by-step explanation:
a, b, and c are the coefficients of the quadratic, so a=1, b=6, and c=-12
(h,k) is the vertex of the parabola, so we must convert it to vertex form by completing the square:
[tex]y=x^2+6x-12\\y+21=x^2+6x-12+21\\y+21=x^2+6x+9\\y+21=(x+3)^2\\y=(x+3)^2-21[/tex]
Therefore, the vertex is (3,-21), which means h=-3 and k=-21.
The y-intercept is when x=0, or when the function passes through the x-axis:
[tex]y=x^2+6x-12\\y=0^2+6(0)-12\\y=-12[/tex]
Therefore, the y-intercept is -12 (also written as (0,-12) as an ordered pair).
The x-intercept is when y=0, or when the function passes through the y-axis:
[tex]\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\x=\frac{-6\pm\sqrt{6^2-4(1)(-12)}}{2(1)}\\\\x=\frac{-6\pm\sqrt{36+48}}{2}\\\\x=\frac{-6\pm\sqrt{94}}{2}[/tex]
Therefore, the x-intercepts are (-6+√94)/2 and (-6-√94)/2.
Hope this helped!
QUESTION 1 1.1 Convert the following to the appropriate units. 1.1.1 1765m to cm 1.1.2 9.4 km to cm 1.1.3 4.2 tons to grams
Answers
Answer:
1. 176,500 centimeters
2. 940,000 centimeters
3.42,000,000 grams
Step-by-step explanation:
Convertion:
1.1.1
1765m to cm
1 meter = 100 centimeters
Therefore, 1765 meters =1765*100 centimeters = 176,500 centimeters
1.1.2
9.4 km to cm
1 kilometer = 1000 meters
First convert it into m.
9.4 kilometers = 9.4*1000 meters = 9400 meters
then in cm
1 meter = 100 centimeters
Therefore, 9400 meters = 9400*100 centimeters = 940,000 centimeters
1.1.3
4.2 tons to grams
1 ton = 1000 kilograms
First convert it into kilogram
1 kilogram = 1000 grams
Therefore, 4.2 tons = 4.2*1000 kilograms = 4200 kilograms
then in grams
4200 kilograms = 4200*1000 grams = 42,000,000 grams
I hope this helps! Let me know if you have any other questions.
Fill out the last question FOR 70 POINTS AND BRAINLIST
Answers
N=LW
N being the number of tiles, L being the length, W being the width. Hope this helps
what is 3 to power of 2 -4 to the power of 5=
Answers
Step-by-step explanation:
Unclear what you are asking
here is one possibility
3^2 - 4^5
9 - 1024 = - 1015
Here is another possibility
(3^(2-4) )^5 = = (3^(-2))^5 = 3 ^-10 = 1/59049
What is the sale price of a pillow originally priced at 12 dollars
Answers
The sale price of a pillow originally priced at 12 dollars is given as follows:
$9.
How to obtain the sale price?The sale price of a pillow originally priced at 12 dollars is obtained applying the proportions in the context of the problem.
The original price is given as follows:
$12.
The markdown rate for this problem is given as follows:
25%.
Hence the sale price of a pillow originally priced at 12 dollars is given as follows:
0.75 x 12 = $9.
(markdown of 25% means that 75% of the price is paid).
Missing InformationThe complete problem is:
"A pillow was marked down at 25%. What is the sale price of a pillow originally priced at 12 dollars?"
More can be learned about proportions at https://brainly.com/question/24372153
#SPJ1
Heeeeeelp pleaseeeeeeeeee!!!!!!!!!!!!L 3.11.3 Quiz: Comparing and Analyzing Function Types
Question 1 of 10
How would you describe the difference between the graphs of f(x)=2x² and
g(x)=-2x²?
A. g(x) is a reflection of f(x) over the line y = x.
B. g(x) is a reflection of f(x) over the line y = -1.
OC. g(x) is a reflection of f(x) over the y-axis.
D. g(x) is a reflection of f(x) over the x-axis.
Answers
The correct statement regarding the reflection rules in this problem is given as follows:
D. g(x) is a reflection of f(x) over the x-axis.
How to define the reflection?The functions f(x) and g(x) in the context of this problem are defined as follows:
f(x) = 2x².g(x) = -2x².The entire function was multiplied by -1, meaning that the signal of the function was exchanged, hence the function was reflected over the x-axis.
As the function was reflected over the x-axis, the correct option in the context of the problem is given by option D.
More can be learned about a reflection over the y-axis at brainly.com/question/27224272
#SPJ1
An ice cream shop has 24 flavors. Anne wants to buy a three scoop cone with three different flavors. How many cones could she buy if the order of the flavors is important.
Answers
If Anne wants to buy a three-scoop cone with three different flavors, and the order of the flavors is important, she has 24 choices for the first scoop, 23 choices for the second scoop (since it must be different from the first), and 22 choices for the third scoop (since it must be different from the first two).
To calculate the total number of cones she could buy, we multiply the number of choices for each scoop:
Number of cones = 24 * 23 * 22
Number of cones = 12,144
Therefore, Anne could buy 12,144 different cones with three different flavors, considering the order of the flavors is important.
I hope this helps! :)
Find the area of a circle with radius,
r
= 46cm.
Give your answer rounded to 3 SF.
Answers
Answer:
6650 cm³
Step-by-step explanation:
area of a circle = π r², where r is the radius
area = π (46)²
= 2116 π
= 6650 cm³ to 3 significant figures
Find the volume of the solid generated by rotating the figure around the given axis. If
necessary, use 3.14 for and round your answer to the nearest hundredth.
TU
2
5
Answers
The volume of the solid generated by rotating the figure is 62.8 cubic units
How to find the volume of the solid generated by rotating the figureFrom the question, we have the following parameters that can be used in our computation:
The figure
The figure is a rectangle with the following dimensions
Length = 5
Width = 2
When rotated around the given axis, we have a cylinder with the following dimensions
Height = 5
Radius = 2
So, we have
Volume = πr²h
This gives
Volume = 3.14 * 2² * 5
Evaluate
Volume = 62.8
Hence, the volume of the solid generated by rotating the figure is 62.8 cubic units
Read more about volume at
https://brainly.com/question/463363
#SPJ1
A camera has a listed price $859.95 of before tax. If the sales tax rate is 9.5% , find the total cost of the camera with sales tax included.
Round your answer to the nearest cent, as necessary.
Answers
Answer:
$941.65
Step-by-step explanation:
Salex tax:We have to find the sales tax of the camera.
tax rate = 9.5%
listed price = $859.95
[tex]\sf Sales \ tax = tax \ rate * listed \ price[/tex]
= 9.5 % * 859.95
= 0.095 * 859.95
= $81.70
To find the total cost, add the sales tax to the listed price.
Total cost = 859.95 + 81.70
= $941.65
Question 3 Consider the input numbers: 3; 5; 7:9; 11 3.1.1 Write all the output numbers if the rule 2x is applied. (2)
Answers
The required outputs of the given rule are 6, 10, 14, 18, 22.
Given inputs are 3, 5, 7, 9, 11 and the rule is 2x.
To find the outputs by applying the given rule to the inputs.
Let x be the input, then the output is 2x.
That implies,
x = 3 then output = 2× 3 = 6.
x = 5 then output = 2× 5 = 10.
x = 7 then output = 2× 7 = 14.
x = 9 then output = 2× 9 = 18.
x = 11 then output = 2× 11 = 22.
Hence. the required outputs of the given rule are 6, 10, 14, 18, 22.
Learn more about the function rule click here:
https://brainly.com/question/30058327
#SPJ1
Based on the W2 form above, how much money did Jane Doe get to take home after taxes in 2017?
Question 9 options:
$48,500
$6,835
$50,000
$725
Answers
The amount of money that Jane Doe get to take home after taxes in 2017 is A. $48500
How to calculate the amountBased on the W-2 form above, Jane Doe's gross pay was $50,000. She had $6,835 in federal income tax withheld, $725 in state income tax withheld, and $1,000 in FICA taxes withheld. This means that her net pay was $48,500.
Net pay = Gross pay - Total taxes withheld
Net pay = $50,000 - ($6,835 + $725 + $1,000)
Net pay = $48,500
Therefore, amount of money that Jane Doe get to take home after taxes in 2017 is A. $48500
Learn more about income on
https://brainly.com/question/25745683
#SPJ1
Whats the surface area
Answers
The surface area of the prism is S = 36 feet²
Given data ,
Let the surface area of the prism be represented as S
Now , the base of the prism is B
where B = l x w
And , length of prism = l and width of prism is = w
The formula for the surface area of a prism is SA=2B+ph, where B, is the area of the base, p represents the perimeter of the base, and h stands for the height of the prism
Surface Area of the prism = 2B + ph
The area of the triangular prism is A = ph + ( 1/2 ) bh
S = 2 ( 1/2 ) ( 3 x 4 ) + ( 4 x 2 ) + ( 3 x 2 ) + ( 5 x 2 )
On simplifying , we get
S = 12 + 8 + 6 + 10
S = 36 feet²
Hence , the surface area is S = 36 feet²
To learn more about surface area of prism click :
https://brainly.com/question/9183590
#SPJ1
e:
3) Find the value of x if RZ-2x+7 and SW= 5x-23
Answers
Applying the properties of the diagonal of a rectangle, the value of x is calculated as: x = 37.
How to Find the Value of x Using the Properties of the Diagonals of a Rectangle?A rectangle is a quadrilateral that has two equal diagonals which bisect each other, to form equal segment. Rectangle RSTW has two diagonals RT and SW, therefore:
2(RZ) = SW
Given the following:
RZ = 2x + 7
SW = 5x - 23
Plug in the values:
2(2x + 7) = 5x - 23
4x + 14 = 5x - 23
4x - 5x = -14 - 23
x = 37
Learn more about rectangle on:
https://brainly.com/question/26228943
#SPJ1
If you are trying to determine which company has higher salaries, which measure of center should you use?
A table titled, Salaries of Current Employees, listing 6 salaries for 2 companies. Company A salaries are: 45,000, 55,000, 53,000, 51,000, 49,000, and 42,000. Company B salaries are: 54,000, 39,000, 51,000, 52,000, 44,000, and 48,000.
mean median mode
Answers
To determine which company has higher salaries, you can use the mean, median, or mode as measures of center. Let's calculate each of these measures for the given data:
Company A salaries: 45,000, 55,000, 53,000, 51,000, 49,000, and 42,000.
Mean (Average):
To calculate the mean, sum up all the salaries and divide by the total number of salaries.
Mean of Company A salaries = (45,000 + 55,000 + 53,000 + 51,000 + 49,000 + 42,000) / 6 = 295,000 / 6 = 49,167
Median:
To find the median, you arrange the salaries in ascending order and find the middle value. If there is an even number of values, you take the average of the two middle values.
Arranging the Company A salaries in ascending order: 42,000, 45,000, 49,000, 51,000, 53,000, 55,000
Median of Company A salaries = (49,000 + 51,000) / 2 = 50,000
Mode:
The mode represents the value(s) that appear most frequently in the dataset.
In Company A salaries, all the values occur only once, so there is no mode.
Now let's calculate the same measures for Company B salaries:
Company B salaries: 54,000, 39,000, 51,000, 52,000, 44,000, and 48,000.
Mean of Company B salaries = (54,000 + 39,000 + 51,000 + 52,000 + 44,000 + 48,000) / 6 = 288,000 / 6 = 48,000
Arranging the Company B salaries in ascending order: 39,000, 44,000, 48,000, 51,000, 52,000, 54,000
Median of Company B salaries = (48,000 + 51,000) / 2 = 49,500
Mode of Company B salaries = There are no repeating values, so there is no mode.
To summarize:
- Mean of Company A salaries: $49,167
- Median of Company A salaries: $50,000
- Mode of Company A salaries: No mode
- Mean of Company B salaries: $48,000
- Median of Company B salaries: $49,500
- Mode of Company B salaries: No mode
Based on these measures of center, we can see that the mean and median salaries of Company A are slightly higher than those of Company B. However, since there is no mode in either dataset, we cannot consider the mode as a determining factor in this case.
Four tiles lettered M, A, T, and H are facedown on a table. A tile is selected, the letter is recorded, the tile is replaced, and the process is repeated.
What is the theoretical probability of selecting the letter T?
Answers
The theoretical probability of selecting the letter T is 1/4.
What is probability?Probability is the possibility of happening an event.
Given the four letters M, A, T, and H.
Note that the letter T occurs 1 time, therefore, the probability of selecting the letter T is:
[tex]\sf P=\dfrac{1}{4}[/tex]
Hence, the theoretical probability of selecting the letter T is 1/4.
Learn more on probabilities here:
https://brainly.com/question/32117953
Rewrite the function by completing the square.
h (x)=x^2+3x−18
Answers
Answer: (x+6)(x-3)
Step-by-step explanation:
y=x^2+3x-18
(x+6)(x-3 )
Solve for m/MNP ifm/MNO = 132° and m/PNO = 40°.
Answers
Answer:
Step-by-step explanation:
To solve for m/MNP if m/MNO = 132° and m/PNO = 40°, we can use the fact that the sum of the angles in a triangle is 180 degrees.
First, we can find the measure of angle MNO:
m/MNO + m/PNO + m/MNP = 180
132 + 40 + m/MNP = 180
m/MNP = 8
Copy
Therefore, m/MNP = 8 degrees.
I hope this helps! Let me know if you have any other questions.
What is the name of an angle that measures 91°?
Answers
Answer: Obtuse Angle
Step-by-step explanation: A 90 degree angle is a right angle anything larger then a right angle is a obtuse angle.
Answer:
It would be called an obtuse angle
Step-by-step explanation: so you see if it was exactly 90 degrees it would be called a right angle. Anything above 90 (b91 92 93 etc...) degrees would be an obtuse angle and anything below 90 ( 89 88 87 etc..) would be an acute angle except an 180 degree angle would be a straight angle. Hope that helps!
have a wonderful Sunday!
1. At a sale on winter clothing, Cody bought two pairs of gloves and four hats for $43.00. Tor bought two pairs of gloves and two hats for
930.00. Find the prices of the hats and gloves.
(k pL.)
Let x-
Let v=
(pt.)
2. Aaliyah invested $10,000 in two mutual funds.
One of the funds rose 6% in one year, and the other rose 906 in one year. Aaliyah's investment rose a total of S684 in one vear. Find the amount she invested in each fund.
Answers
1. The price of a pair of gloves (x) is $8.50, and the price of a hat (y) is $6.50.
2. Aaliyah invested $7,200 in the mutual fund that rose 6% and $2,800 in the mutual fund that rose 9%.
1. Let's assume the price of a pair of gloves is represented by 'x' and the price of a hat is represented by 'y'.
From the given information, we can establish the following equations based on Cody and Tor's purchases:
Equation 1: 2x + 4y = 43.00 (Cody's purchase)
Equation 2: 2x + 2y = 30.00 (Tor's purchase)
To solve this system of equations, we can use a method like substitution or elimination.
Using the substitution method, we can rearrange Equation 2 to express x in terms of y:
2x = 30.00 - 2y
x = 15.00 - y
Now, we substitute this value of x into Equation 1:
2(15.00 - y) + 4y = 43.00
30.00 - 2y + 4y = 43.00
2y = 43.00 - 30.00
2y = 13.00
y = 6.50
Substituting this value of y back into Equation 2 to find x:
2x + 2(6.50) = 30.00
2x + 13.00 = 30.00
2x = 30.00 - 13.00
2x = 17.00
x = 8.50
Therefore, the price of a pair of gloves (x) is $8.50, and the price of a hat (y) is $6.50.
2. Let's assume Aaliyah invested an amount represented by 'x' in the mutual fund that rose 6% and an amount represented by 'y' in the mutual fund that rose 9%.
From the given information, we can establish the following equations based on the rise in investment:
Equation 1: 0.06x + 0.09y = 684 (rise in investment)
Equation 2: x + y = 10,000 (total investment)
To solve this system of equations, we can use a method like substitution or elimination.
Using the substitution method, we can rearrange Equation 2 to express x in terms of y:
x = 10,000 - y
Now, we substitute this value of x into Equation 1:
0.06(10,000 - y) + 0.09y = 684
600 - 0.06y + 0.09y = 684
0.03y = 84
y = 2800
Substituting this value of y back into Equation 2 to find x:
x + 2800 = 10,000
x = 10,000 - 2800
x = 7200
Therefore, Aaliyah invested $7,200 in the mutual fund that rose 6% and $2,800 in the mutual fund that rose 9%.
for such more question on price
https://brainly.com/question/30036691
#SPJ8
Find the probability that
event A or B takes place.
A
B
10
19
19
Answers
The probability that event A or B takes place is 17/19
Probability of an eventProbability is defined as the ratio of the required to the total possible outcomes of an event.
The probability of event A or B taking place can be defined as ;
P(A or B) = P(A) + P(B)
P(A) = 10/19
P(B) = 7/19
Hence,
P(A) + P(B) = 10/19 + 7/19
L.C.M of 19 and 19 = 1
P(A) + P(B) = (10 + 7)/19
P(A) + P(B) = 17/19
Therefore, the probability that event A or B takes place is 17/19
Learn more on probability ;https://brainly.com/question/13604758
#SPJ1
Please solve this
∫ (log(1 + x ^ 2))/((x + 1) ^ 2) dx
Answers
The final result of the integral is:
∫ (log(1 + x²) / (x + 1)²) dx = log(x + 1) - 2 (log(x + 1) / x) - 2Li(x) + C,
where Li(x) is the logarithmic integral function and C is the constant of integration.
We have,
To solve the integral ∫ (log(1 + x²) / (x + 1)²) dx, we can use the method of substitution.
Let's substitute u = x + 1, which implies du = dx. Making this substitution, the integral becomes:
∫ (log(1 + (u-1)²) / u²) du.
Expanding the numerator, we have:
∫ (log(1 + u² - 2u + 1) / u²) du
= ∫ (log(u² - 2u + 2) / u²) du.
Now, let's split the logarithm using the properties of logarithms:
∫ (log(u² - 2u + 2) - log(u²)) / u² du
= ∫ (log(u² - 2u + 2) / u²) du - ∫ (log(u²) / u²) du.
We can simplify the second integral:
∫ (log(u²) / u²) du = ∫ (2 log(u) / u²) du.
Using the power rule for integration, we can integrate both terms:
∫ (log(u² - 2u + 2) / u²) du = log(u² - 2u + 2) / u - 2 ∫ (log(u) / u³) du.
Now, let's focus on the second integral:
∫ (log(u) / u³) du.
This integral does not have a simple closed-form solution in terms of elementary functions.
It can be expressed in terms of a special function called the logarithmic integral, denoted as Li(x).
Therefore,
The final result of the integral is:
∫ (log(1 + x²) / (x + 1)²) dx = log(x + 1) - 2 (log(x + 1) / x) - 2Li(x) + C,
where Li(x) is the logarithmic integral function and C is the constant of integration.
Learn more about integrations here:
https://brainly.com/question/30217024
#SPJ1