1. (a) Evaluate the following integrals (i) √ x√2x² −5 dx x cos 2x dx (ii) x+1 (iii) dx (x+2)(x+3) (3 Marks)
Answers
Evaluating the integrals results to
∫√(x√2x² - 5) dx = \(\sqrt{2} (x^{2} \sqrt{2} - 5) ^{1/2}\) + C.
∫x cos(2x) dx = (1/2) x sin(2x) + (1/4) cos(2x) + C
∫ dx / ((x+2)(x+3)) = (1/5) ln|x+3| - (1/5) ln|x+2| + C.
How to evaluate the integralsevaluating the given integrals one by one:
(i) ∫√(x√2x² - 5) dx:
∫√(x√2x² - 5) dx = ∫√(x * x√2 - 5) dx = ∫√(x²√2 - 5) dx.
if u = x²√2 - 5.
du/dx = 2x√2,
dx = du / (2x√2)
substituting these values into the integral:
∫√(x²√2 - 5) dx = ∫√u * (du / (2x√2)) = (1 / (2√2)) ∫√u / x du.
factoring out \(u^{1/2}\) / x, we get:
(1 / (2√2)) ∫(\(u^{1/2}\) / x) du
= (1 / (2√2)) ∫\(u^{1/2}\) * u⁻¹ du
= (1 / (2√2)) ∫\(u^{-1/2}\) du.
Integrating \(u^{-1/2}\)
(1 / (2√2)) * (2\(u^{1/2}\)) + C = √2\(u^{1/2}\) + C,
where C is the constant of integration.
Finally, substitute back u = x²√2 - 5 to get the final result:
∫√(x√2x² - 5) dx = \(\sqrt{2} (x^{2} \sqrt{2} - 5) ^{1/2}\) + C.
(ii) ∫x cos(2x) dx:
To evaluate this integral, we can use integration by parts.
if u = x and dv = cos(2x) dx.
du = dx and v = (1/2)sin(2x).
Using the integration by parts formula ∫u dv = uv - ∫v du, we can write:
∫x cos(2x) dx = (1/2)x sin(2x) - (1/2)∫sin(2x) dx.
Integrating sin(2x)
(1/2)x sin(2x) + (1/4)cos(2x) + C,
(iii) ∫dx / ((x+2)(x+3))
To evaluate the integral ∫ dx / ((x+2)(x+3)), we can use partial fraction decomposition.
∫ dx / ((x+2)(x+3)) = ∫ (A/(x+2) + B/(x+3)) dx.
multiplying both sides by (x+2)(x+3)
1 = A(x+3) + B(x+2).
Expanding and equating coefficients
1 = (A + B)x + (3A + 2B).
A + B = 0 and 3A + 2B = 1.
Solving these equations, we find A = -1/5 and B = 1/5.
Substituting the values of A and B back into the integral, we have:
∫ dx / ((x + 2) (x + 3)) = ∫ (-1/5(x + 2) + 1/5(x + 3)) dx,
= (-1/5) ln |x + 2| + (1/5) ln |x + 3| + C,
= (1/5) ln |x + 3| - (1/5) ln |x + 2| + C.
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Related Questions
Divide.
−313÷9
what is it?
Answers
Answer:
-34.777777777778
Step-by-step explanation:
Answer:
-34.7 with a line above the 7
Step-by-step explanation:
let d be the solid region bounded by the paraboloids and . write six different triple iterated integrals for the volume of d. evaluate one of the integrals.
Answers
To find the volume of the solid region bounded by the paraboloids y = x^2 and z = 4 - x^2, we need to set up triple iterated integrals in terms of x, y, and z.
One way to do this is to integrate over x first, then y, then z, or vice versa. Here are six different triple iterated integrals we can use:
1. ∫∫∫d dz dy dx
2. ∫∫∫d dx dy dz
3. ∫∫∫d dx dz dy
4. ∫∫∫d dy dx dz
5. ∫∫∫d dy dz dx
6. ∫∫∫d dz dx dy
Let's evaluate the first integral:
∫∫∫d dz dy dx
We start by finding the limits of integration for z. The paraboloid z = 4 - x^2 is above the paraboloid y = x^2, so the lower limit for z is y - x^2, and the upper limit is 4 - x^2.
Next, we find the limits of integration for y. The paraboloid y = x^2 is a function of x, so the limits are given by the x-values that bound the region d. Since the paraboloids intersect at x = -2 and x = 2, the limits for y are x^2 and 4 - x^2.
Finally, we find the limits of integration for x. The region d is symmetric about the yz-plane, so we can integrate over x from 0 to 2 and multiply by 2 to get the full volume. Therefore, the limits for x are 0 and 2.
Putting it all together, we have:
∫∫∫d dz dy dx = ∫0^2 ∫x^2^(4-x^2) ∫y-x^2^(4-x^2) dz dy dx
Evaluating this integral is a bit messy, but it can be done with some algebraic manipulation and trigonometric substitutions. The answer turns out to be: 64/15
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which statement about the graph of the line y=x+1/2 will be true if the 1/2 is replaced by -2
Answers
Answer:
Step-by-step explanation:
it help me alot
Write each number as a percent.
1(2/5)
Answers
These are Helen’s top three financial goals: create an emergency fund, save for a new guitar, and invest in her company’s stock. Select one attribute that matches each goal. Goal Attribute Create an emergency fund by next fall. specific, timely Save $50 per month to purchase a new guitar. timely, measurable Invest $130 in company stock. specific, timely
Answers
Answer:
1. timely
2. measurable
3. specific
Step-by-step explanation:
Shawn rents an apartment in an all-brick building located in a historic,
downtown neighborhood. The value of the belongings in the apartment is
about $25,000. If Shawn wishes to insure his belongings while renting, how
much will he have to pay for insurance per year?
Answers
Answer: 107.50
Step-by-step explanation:
RCV is more expensive compared to ACV, therefore, Shawn would have to pay more to insure his belongings with RCV coverage.
As Shawn rents an apartment in an all-brick building located in a historic, downtown neighborhood and the value of the belongings in the apartment is about $25,000, he will have to pay around $125 to $375 per year to insure his belongings while renting depending on the type of coverage he chooses.
A renter’s insurance policy can protect the belongings of renters who live in an apartment or house they rent and covers their liability towards others in case they get injured. For this reason, the cost of renter’s insurance will vary depending on the type of coverage needed. If Shawn wants more coverage, then he will pay a higher rate. There are two types of coverage which include actual cash value (ACV) and replacement cost value (RCV).
ACV is a coverage that pays for the value of the belongings, minus the depreciation. Whereas, RCV is a coverage that pays for the cost of replacing the belongings, regardless of their current value.
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What is a similarity transformation in geometry?
Answers
A similarity transformation is a rigid movement followed by scaling, so this type of transformation can change the position and size, but not the shape.
In geometry, a similarity transformation is a rigid movement followed by scaling, so this type of transformation can change the position and size, but preserve the shape and angles.
Thus, two or more shapes are similar if they have the same shape and corresponding angles, but their corresponding sides are proportional in length.
This type of transformation is often also referred to as dilation.
The rule of dilatation:
\(D_{k}\) (x,y) = (kx, ky)
with \(D_{k}\) is the coordinate point of the new image and k is the scale factor.
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riddle answer i am a number i am not a odd number but i am higer than 90 but not abouve 100 what number am i
Answers
The number is 100.
The method for identifying or representing numbers is known as the number system or numeral system. We are aware that a number is a mathematical value that aids in the measurement and counting of items as well as in several mathematical operations.
The number is not an odd number and it's higher than 90 but Not higher than 100.
So, taking even numbers from 90 to 100.
90
92
94
96
98
100
Given: If the number is subtracted from 100 we get nothing which means we get "0" as the Answer.
So, 100 - 100 = 0. (zero has No value)
Hence: the number is 100.
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two hundred and seven thousand in standard form
Answers
Answer:
7,200
Step-by-step explanation:
I hope this helps!
Plsssss help me I will give brainiest
Answers
Answer:
the first one: 1:39
the second one: 7:12
the third one: 10:24
Step-by-step explanation:
the short hand points to the hour and the long hand points to the minutes. when the long hand is at 1, that's 5 minutes and when it's at 2, that's 10 minutes and so on and so forth. and each of the little dash marks in between the numbers is 1 minute.
i hope this was correct, i'm not great with time
Question 1:
say 1:39am/pm OR one thirty-nineQuestion 2:
say 7:12am/pm OR seven twelveQuestion 3:
say 10:24am/pm OR ten twenty-fourname the multiplicative inverse of 3/7
Answers
Answer:The multiplicative inverse of a number is the number by which when we multiply that number then final result is 1. Let , multiplicative inverse of ( -3/7 ) is x
Step-by-step explanation:
Find the difference between points M(6, 16) and Z(-1, 14) to the nearest tenth.
Answers
The distance between points M(6, 16) and Z(-1, 14) is approximately 7.1 units.
We can use the distance formula to find the distance between two points in a coordinate plane. The distance formula is:
d = √((x2 - x1)² + (y2 - y1)²)
where (x1, y1) and (x2, y2) are the coordinates of the two points. Substituting the coordinates of M(6, 16) and Z(-1, 14), we get:
d = √((-1 - 6)² + (14 - 16)²) = √(49 + 4) = √53 ≈ 7.1
Therefore, the distance between points M(6, 16) and Z(-1, 14) is approximately 7.1 units.
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Which statement is NOT true concerning the graphing of a linear system?
You can only graph a system of linear equations with a graphing calculator.
Systems of linear equations that contain coordinates with fractions or decimals can be challenging.
You need to be careful and plot straight lines.
The point of intersection of lines that intersect at a shallow angle is difficult to determine.
Answers
Answer:
1
Step-by-step explanation:
the first one it's not right
The statement which is NOT true concerning the graphing of a linear system is You can only graph a system of linear equations with a graphing calculator.
What is graphing of linear system?Every linear equation in two variables can be represented geometrically as a straight line in a coordinate plane. Points on the line are the solution of the equation. This why equations with degree one are called as linear equations.
Graphing of a linear system includes :
Recognize the relationship between the solutions of an equation and its graph.Graph a linear equation by plotting points.Graph vertical and horizontal linesAccording to the question
Graphing of a linear system includes :
Systems of linear equations that contain coordinates with fractions or decimals can be challenging.You need to be careful and plot straight lines.The point of intersection of lines that intersect at a shallow angle is difficult to determine.But NOT true for :
You can only graph a system of linear equations with a graphing calculator.
Hence, the statement which is NOT true concerning the graphing of a linear system is You can only graph a system of linear equations with a graphing calculator.
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FAST pls will give brainliest
Answers
Answer:
D
Step-by-step explanation:
Last time I took the test I got it right
Which statement compares the two numbers correctly?
6x
+3x
+2x
100
six hundred twenty-nine thousandths
1000
06x
6x+3x) +2xਰ)
six hundred twenty-nine thousandths
1
O।
+2
= six hundred twenty-nine thousandths
100
six hundred twenty-nine thousandths < 6x
( 6 x+2+ਰ ) + 2 ਅਰਹਰ)
> 6 x+2+ਚ) • ਅਰ)
O six hundred twenty-nine thousandths > 6x
Answers
Answer:
1
Step-by-step explanation:
(Giving brainilest) For the data set 62, 73,74,75,76, the mean is 72. what is the mean absoulute deviation
Answers
Answer:
C. The mean absolute deviation is 4.
Step-by-step explanation:
It is asking how far away is the mean from the farthest number. That is what deviation means. 72 is 4 numbers away from both 62 and 76.
Answer:
C, 4
Step-by-step explanation:
To find mean absolute deviation we have to find the difference of each data point with the mean of 72
1. |72-62| = 10
2. |72-73|=|-1|=1
3. |72-74| =|-2|=2
4. |72-75|= |-3|= 3
5. |72-76|= |-4| =4
Now we vind the average of these differences by adding them all up and dividing by the number there are:
\(\frac{10+1+2+3+4}{5} = \frac{20}{5} =4\)
C
I hope this helps! :)
A shirt costing $200 is discounted 15%. After a month, the shirt is discounted another 15%. Which of the following expressions can be used to find the selling price of the shirt? a. (200)(0.70) b. (200)-(.30)(200) C. (200)(0.15) -(200)(0.15) d. (200)(0.85)(0.85)
Answers
The expression that can be used to find the selling price of the shirt is option d, which is (200)(0.85)(0.85).
This expression takes into account both discounts that the shirt undergoes, where the first discount reduces the price by 15%, leaving 85% of the original price. After a month, the discounted price is then reduced by another 15%, which means we need to multiply the remaining 85% by another 85%.
To see why option d is correct, we can break down the calculation step-by-step. The first discount reduces the price of the shirt to 85% of the original price, which can be computed as:
(200)(0.85) = 170
This means the discounted price of the shirt is $170. After a month, the discounted price is further reduced by another 15%, which means we need to multiply the remaining 85% by another 85%. We can compute this as:
(170)(0.85) = 144.50
Therefore, the selling price of the shirt is $144.50, which matches the result obtained by evaluating option d.
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Johnny has 6 /4 feet of material for his costume. He needs 10 2/3 feet for the entire costume. How much more material does he need?
Answers
Answer:
9 2/12 more feet
Step-by-step explanation:
this take over causes the cell to _____________ instead of makeing new cells
Answers
Meiosis causes the cell to divide instead of makeing new cells.
What are cells?Cells are the fundamental building blocks of all life. Trillions of cells make up the human body. They support the body's structure, absorb nutrients from food, convert those nutrients into energy, and perform specialized functions.
Meiosis is a type of cell division that occurs in sexually reproducing organisms and results in a reduction in the number of chromosomes in gametes; he sex cells, or egg and sperm. Body or somatic cells in humans are diploid, with two sets of chromosomes one from each parent.
It should be noted that meiosis is a type of cell division of germ cells that produces gametes such as sperm or egg cells in sexually reproducing organisms. It consists of two rounds of division that result in four cells with only one copy of each chromosome.
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23. Jeremy is two years older than Rachel. The sum of the ages of Jeremy and Rachel is less
How old could Jeremy be?
Let
Inequality:
Answers
The age of Jeremy could be less than 22 years
How to determine how old Jeremy could be?From the question, we have the following parameters that can be used in our computation:
Jeremy = Rachel - 2
Let Jeremy = x and Rachel = y
So, we have
y = y - 2
Their ages added together is less than 46
So, we have
x + y < 46
This gives
x + x + 2 < 46
So, we have
2x < 44
Divide
x < 22
Hence, Jeremy could be less than 22 years
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Question
Jeremy is two years older than Rachel. The sum of the ages of Jeremy and Rachel is less than 46
How old could Jeremy be?
The following sequence is an arithmetic sequence. [-9, -2, 5, 12,...]
Find the common difference and the next two terms.
![The following sequence is an arithmetic sequence. [-9, -2, 5, 12,...]Find the common difference and the](https://i5t5.c14.e2-1.dev/h-images-qa/contents/attachments/adr8CLuxqNUEu1GVh8jKvzIp0ivI0p5O.png)
Answers
Answer: The common difference is 7, and the next two terms are 19 and 26.
Step-by-step explanation:
Between -9 and -2 there is +7, same with between -2 and 5, as well as 5 and 12. Adding 7 to 12 again gets 19, and adding 7 to 19 gets 26 as well for the next two terms.
complete each
1. 570 mL = _______________ L
2. 623 cm3 = _______________ m3
3. 38 gal. = _______________c
4. 585mL = ________________tbsp
5. 20 fl. oz. = ________________ tsp
6. 12 hrs =________________ min
Answers
Answer:
1. 570 mL = 0.57 L
2. 623 cm3 = 0.000623 m3
3. 38 gal. = 608 cups
4. 585mL = 39.5624 tbsp
5. 20 fl. oz. = 120 tsp
6. 12 hrs = 720 min
Step-by-step explanation:
1. 570 mL to L
divide the volume value by 1000, 570 / 1000 = 0.57
2. 623 cm3 to m3
divide the volume value by 623/10^6 = 0.000623
3. 38 gal. to c
1 gallon is 16 cups,
multiply the volume value by 16, 38 x 16 = 608c
4. 585mL to tbsp
for an approximate result, divide the volume value by 14.787
14.787 x 585 = 39.5624
5. 20 fl. oz. to tsp
multiply the volume value by 6, 20 x 6 = 120
6. 12 hrs to min
1 hour is 60 minutes
multiply the time value by 60
12 x 60 = 720
Find the measure of line BD.
Answers
Answer:
Step-by-step explanation:
BD is geometric mean
BD² = 5 × 13 = 65
BD = √65
On a bike trip across the untied states, rodney notes that he covers about 190 miles every 4 days. if he continues at this rate, how many miles he could bike in 6 days
Answers
Answer:
285 miles
Step-by-step explanation:
190 miles i 4 days (divide by 4)
47.5 miles in 1 day (multiply by 6)
285 miles in 6 days
For the key assumption of normal distribution for multiple
linear regression analysis, what is the problem if they are not
normally distributed?
Answers
Adherence to the assumption of normality is crucial for obtaining valid and meaningful results in multiple linear regression analysis. It affects the validity of the statistical inference, making it difficult to interpret the significance of the estimated coefficients and their corresponding p-values.
1. The assumption of normal distribution in multiple linear regression analysis is essential for several reasons. When the errors or residuals (the differences between the observed and predicted values) are normally distributed, it allows for the validity of statistical inference. This means that the estimated coefficients and their associated p-values accurately reflect the relationships between the independent variables and the dependent variable in the population.
2. When the assumption of normality is violated, it can lead to problems with statistical inference. Non-normal errors can result in biased coefficient estimates, making it difficult to interpret the true relationships between the variables. Additionally, the p-values obtained for the coefficients may be inaccurate, potentially leading to incorrect conclusions about their significance.
3. Moreover, non-normality can distort the predictions made by the regression model. In a normally distributed error term, the predicted values are unbiased estimators of the true values. However, if the errors are not normally distributed, the predictions may be systematically overestimated or underestimated, leading to unreliable forecasts.
4. To address this issue, several techniques can be employed. One approach is to transform the variables to achieve approximate normality, such as using logarithmic or power transformations. Alternatively, robust regression methods that are less sensitive to deviations from normality can be utilized. It is also important to consider the underlying reasons for the non-normality, such as outliers or influential observations, and address them appropriately.
5. In conclusion, adherence to the assumption of normality is crucial for valid and meaningful results in multiple linear regression analysis. Violations of this assumption can affect the statistical inference and prediction accuracy, highlighting the importance of assessing and addressing normality in the data.
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Regina spent $45.50 over 612 months on music downloads.
If she spent the same amount of money each month, how much money did Regina spend per month on music downloads?
Answers
Answer:
.07 cents
Step-by-step explanation:
In order to solve this question all you need to do is divide the money spent by the amount of months. 45.50/612=0.0743464052 but since you cant spend .0043464052 of a cent we have to change the answer to .07. To check our answer is right all we need to do is multiply 0.0743464052 by 612=45.499999 so we know the answer is correct.
A Girl Scout camp served half of their granola with breakfast. After dinner, they put the remaining 6{,}250 \text{ g}6,250 g6, comma, 250, start text, space, g, end text of granola on top of their yogurt dessert. How many kilograms of granola did the Girl Scout camp start with?
Answers
Answer:
12.5 kilograms
Step-by-step explanation:
Given:
Half of the granola served with breakfast.
Let the amount of Granola present initially = \(x\) kg
Half of it (\(\frac{x}{2}\)) served with breakfast, so remaining granola :
\(x-\dfrac{x}{2} = \dfrac{x}{2}\)
After dinner, the remaining granola 6,250 gm granola was served with yogurt dessert.
First of all, let us convert it to kilogram because the answer is required in Kilograms.
We know that:
1000 gms = 1 kilograms
1 gms = 0.001 kilograms
6250 gms = 6250 \(\times\) 0.001 = 6.250 kilograms
Now, as per given question statement:
\(\dfrac{x}{2} = 6.250\ kg\\\Rightarrow x = 2 \times 6.250\\\Rightarrow \bold{x = 12.5\ kg}\)
So, the answer is:
Girl scout camp started with 12.5 Kilograms of granola.
A car company tested a sports car on a road with different inclines. the test driver tested the car by driving a distance of x miles on a flat road, (x2 3) miles downhill, and (x − 7) miles uphill. which simplified expression is equivalent to the total distance, in miles, for which the car was tested? 3x2 − 4 3x2 10 x2 2x − 4 x2 2x 10
Answers
Total distance represented by the simplified expression for which the car was tested equals option c. x² + 2x - 4 .
Total distance drive on flat road = x miles
Total distance drive on downhill = (x²+ 3) miles
Total distance drive on uphill = ( x - 7 ) miles
Simplified expression which is equivalent to total distance drive by a car
= Distance drive on ( flat road + downhill + uphill )
= ( x + x²+ 3 + x - 7 ) miles
= ( x² + 2x - 4 ) miles
Therefore, the simplified expression equivalent to the total distance drive by a tested car is equal to option c. ( x² + 2x - 4 ) miles.
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The above question is incomplete , the complete question is:
A car company tested a sports car on a road with different inclines. The test driver tested the car by driving a distance of x miles on a flat road, (x²+ 3) miles downhill, and (x - 7) miles uphill. Which simplified expression is equivalent to the total distance, in miles, for which the car was tested?
a.3x² _ 4
b. 3x² + 10
c. x² + 2x - 4
d. x² + 2x + 10
SIMPLIFY!!!
2.8d + 3.6d
Answers
Answer:
6.4d
Step-by-step explanation:
Simple as that. d is like an x
Look over Chuck's work What is incorrect about the way Chuck interpreted his problem? What should have been a clue to Chuck that something was wrong?
Answers
The probability that a random student will be taking both Algebra 2 and Chemistry is 0.0136 or 1.36%.
To find the probability that a random student will be taking both Algebra 2 and Chemistry, we need to use the concept of conditional probability.
Let's denote the event of taking Algebra 2 as A and the event of taking Chemistry as C. We are given that P(A) = 0.08 (8% probability of taking Algebra 2) and P(C|A) = 0.17 (17% probability of taking Chemistry given that the student is taking Algebra 2).
The probability of taking both Algebra 2 and Chemistry can be calculated using the formula for conditional probability:
P(A and C) = P(C|A) * P(A)
Substituting the given values:
P(A and C) = 0.17 * 0.08
P(A and C) = 0.0136
Therefore, the probability that a random student will be taking both Algebra 2 and Chemistry is 0.0136 or 1.36%.
It is important to note that the probability of taking both Algebra 2 and Chemistry is determined by the intersection of the two events, which means students who are taking both courses. In this case, the probability is relatively low, as it depends on the individual probabilities of each course and the conditional probability given that a student is taking Algebra 2.
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