Evaluate the integral. (Remember to use In |u| where appropriate. Use C for the constant of integration.) x2 - x + 12 / x3 + 3x dx Find the area of the region under the given curve from 1 to 5. y = X2 + 7 / 6X - X2 Make a substitution to express the integrand as a rational function and then evaluate the integral. (Remember to use In |u| where appropriate. Use C for the constant of integration.) ex / (ex / (ex - 7)(e2x + 1)dx)
Answers
First, let's evaluate the integral:
We can make a substitution by letting u = x^3 + 3x, which means du/dx = 3x^2 + 3. This can be rewritten as (1/3)(3x^2 + 3) dx = x^2 + 1 dx, which means dx = (1/(x^2 + 1)) du/3.
Substituting this into the integral, we get:
∫(x^2 - x + 12)/(x^3 + 3x) dx = ∫(x^2 - x + 12)/(x(x^2 + 3)) dx
= ∫(1/3)((u/x^3) - (u/x) + (4/x(x^2 + 3))) du
Now, we can integrate each term separately:
= (1/3) ln |u| - (1/3) ln |x| + (2/3) ln |x^2 + 3| + C
Substituting back in for u, we get:
= (1/3) ln |x^3 + 3x| - (1/3) ln |x| + (2/3) ln |x^2 + 3| + C
Next, let's find the area of the region under the curve y = (x^2 + 7)/(6x - x^2) from x = 1 to x = 5:
We can first find the antiderivative of the integrand by making the substitution u = 6x - x^2, which means du/dx = 6 - 2x. This can be rewritten as (-1/2)(6 - 2x) dx = (x - 3) dx, which means dx = (1/(x-3)) du/(-2).
Substituting this into the integral, we get:
∫(x^2 + 7)/(6x - x^2) dx = ∫-(1/2)((u/(u-36)) du
Now, we can integrate:
= (1/2) ln |u| - (1/2) ln |u-36| + C
Substituting back in for u, we get:
= (1/2) ln |6x - x^2| - (1/2) ln |6x - x^2 - 36| + C
To find the area of the region under the curve, we need to evaluate this antiderivative from x = 1 to x = 5 and take the absolute value of the result:
|∫(x^2 + 7)/(6x - x^2) dx from x = 1 to x = 5|
= |((1/2) ln |30 - 25| - (1/2) ln |30 - 61|) - ((1/2) ln |6 - 1| - (1/2) ln |6 - 36|)|
= |(1/2) ln (5/31) + (1/2) ln (5/30)|
= (1/2) ln (1/31)
Therefore, the area of the region under the curve from x = 1 to x = 5 is (1/2) ln (1/31).
Finally, let's evaluate the integral ∫(e^x)/((e^x - 7)(e^(2x) + 1)) dx:
We can make the substitution u = e^x, which means du/dx = e^x.
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Related Questions
multistage cluster sampling with stratification, systematic sampling, simple random sampling and disproportionate stratified sampling are examples of:
Answers
Multistage cluster sampling with stratification, systematic sampling, simple random sampling and disproportionate stratified sampling are examples of probability sampling.Probability sampling is the process of selecting a sample from a population when the selection is based on the randomization principle, often known as chance or random selection.
What does stratified multistage cluster sampling mean?Multistage sampling, also known as multistage cluster sampling, involves taking a sample from a population in successively smaller groupings. In national surveys, for instance, this technique is frequently employed to collect data from a sizable, geographically dispersed population.For instance, a researcher might be interested in the various eating customs throughout western Europe. It is essentially impossible to gather information from every home. The researcher will first pick the target nations. He or she selects the states or regions to survey from among these nations.
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Important Please Help!!!!!!
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A pendant hanging on a necklace is in the shape of a rhombus. The front of the pendant is covered with gold leaf.
A rhombus where the left side of the horizontal diagonal has length 7.5 millimeters and the right side of the horizontal diagonal has length 7.5 millimeters. The top of the vertical diagonal has length 11 millimeters and the bottom of the vertical diagonal has length 11 millimeters.
How much gold leaf is needed to cover the pendant?
18.5 mm2
82.5 mm2
165 mm2
330 mm2
Answers
please give brainliest
Answer:
To find the amount of gold leaf needed to cover the pendant, we can calculate the area of the rhombus. The area of a rhombus is given by the formula: area = (diagonal 1 * diagonal 2) / 2.
In this case, the diagonals are given as follows:
Left side of the horizontal diagonal: 7.5 mm
Right side of the horizontal diagonal: 7.5 mm
Top of the vertical diagonal: 11 mm
Bottom of the vertical diagonal: 11 mm
Now, we can calculate the area:
Area = (7.5 mm * 11 mm) / 2
Area = 82.5 mm²
Therefore, the amount of gold leaf needed to cover the pendant is approximately 82.5 mm².
a ball of radius 13 has a round hole of radius 3 drilled through its center. find the volume of the resulting solid.
Answers
a ball of radius 13 has a round hole of radius 3 drilled through its center. then the volume of the resulting solid is 8.5528 x 10³ units³
The radius of the ball is 13 units.
Its volume is
V₁ = (4π*13³)/3
= 9.2028 x 10³ units³
The drilled hole has the approximate shape of a solid cylinder with radius = 3 and length = 26. Its volume is
V₂ = π*(3²)*26
= 0.650 x 10³ units³
The volume of the resulting solid is
V₁ - V₂ = 8.5528 x 10³
Answer: 8.5528 x 10³ units³
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If f (x) = x2 + 3x – 2 , find f (-2).
PLEASE HELP!
Answers
Answer:
f(- 2) = - 4
Step-by-step explanation:
Step-by-step explanation:
To evaluate f(- 2) substitute x = - 2 into f(x) , that is
f(- 2) = (- 2)² + 3(- 2) - 2 = 4 - 6 - 2 = - 4
The coordinate grid shows points A through K. What point is a solution to the system of inequalities? y > −2x + 10 y > 1/2x − 2 coordinate grid with plotted ordered pairs, point A at negative 5, 4 point B at 4, 7 point C at negative 2, 7 point D at negative 7, 1 point E at 4, negative 2 point F at 1, negative 6 point G at negative 3, negative 10 point H at negative 4, negative 4 point I at 9, 3 point J at 7, negative 4 and point K at 2, 3
answer:
a) E
b) K
c) B
d) D
Answers
Based on the analysis, point D at (-7, 1) is the only solution to the system of inequalities y > -2x + 10 and y > (1/2)x - 2. Therefore, the correct answer is option d) D.
To determine which point is a solution to the system of inequalities y > -2x + 10 and y > (1/2)x - 2, we can test each point to see if it satisfies both inequalities.
a) Point E at (4, -2):
Substituting the coordinates into the inequalities:
-2 > -2(4) + 10 -> -2 > -8 + 10 -> -2 > 2 (False)
-2 > (1/2)(4) - 2 -> -2 > 2 - 2 -> -2 > 0 (False)
b) Point K at (2, 3):
Substituting the coordinates into the inequalities:
3 > -2(2) + 10 -> 3 > -4 + 10 -> 3 > 6 (False)
3 > (1/2)(2) - 2 -> 3 > 1 - 2 -> 3 > -1 (True)
c) Point B at (4, 7):
Substituting the coordinates into the inequalities:
7 > -2(4) + 10 -> 7 > -8 + 10 -> 7 > 2 (True)
7 > (1/2)(4) - 2 -> 7 > 2 - 2 -> 7 > 0 (True)
d) Point D at (-7, 1):
Substituting the coordinates into the inequalities:
1 > -2(-7) + 10 -> 1 > 14 + 10 -> 1 > 24 (False)
1 > (1/2)(-7) - 2 -> 1 > -3.5 - 2 -> 1 > -5.5 (True)
Based on the analysis, point D at (-7, 1) is the only solution to the system of inequalities y > -2x + 10 and y > (1/2)x - 2. Therefore, the correct answer is option d) D.
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hundred and thous
d. 39)87 652
B
Answers
Write an expression to represent the area of a right triangle with legs length as follow: 2x-2 and 4x+2
Answers
A=ab/2
(2x-2)(4x+2)/2
(8x^2+4x-8x-4)/2
(8x^2-4x-4)/2 take common term 2
2(4x^2-2x-2)/2
So A=4x^2-2x-2
Expression to represent the area of a right triangle with legs length as 2x-2 and 4x+2 is 4x² - 2x - 2.
What is expression?Expressions in math are mathematical statements that have a minimum of two terms containing numbers or variables, or both, connected by an operator in between. The mathematical operators can be of addition, subtraction, multiplication, or division.
For example, x + y is an expression, where x and y are terms having an addition operator in between.
Given,
legs length of a right angle triangle = 2x-2 and 4x+2
Area of the right angle triangle = (length of leg a × length of leg b)/2
= (2x-2)(4x + 2)/2
= 2(x - 1)(4x + 2)/2
= (x - 1)(4x + 2)
Area of the right angle triangle = 4x² - 2x - 2
Hence, 4x² - 2x - 2 is the expression for area of right triangle.
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which sampling method is being described
A store manger randomly choose a shopper entering her store to interview she then interview every 20th person after that contomer
to do the survey
Answers
Systematic sampling offers several advantages. It is relatively easy to implement and eliminates bias that may arise from the subjective selection of participants.
The sampling method described in the scenario is called systematic sampling.
Systematic sampling involves selecting every nth element from a population after randomly selecting a starting point. In this case, the store manager randomly chooses a shopper entering the store as the starting point and then proceeds to interview every 20th person after that initial selection.
Systematic sampling offers several advantages. It is relatively easy to implement and eliminates bias that may arise from the subjective selection of participants. By ensuring a regular interval between selections, systematic sampling provides a representative sample from the population.
However, it's important to note that systematic sampling can introduce a form of bias if there is any periodicity or pattern in the population. For example, if the store experiences a peak in customer traffic during specific time periods, the systematic sampling method might overrepresent or underrepresent certain groups of shoppers.
To minimize this potential bias, the store manager could randomly select the starting point for the systematic sampling at different times of the day or on different days of the week. This would help ensure a more representative sample and reduce the impact of any inherent patterns or periodicities in customer behavior.
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Geometry area please help find the shaded
Answers
The shaded area is 239.4 ft²
Determine the complex exponential Fourier series representation for each of the following signals: X(t) = cos wt x(t) = -cos(0.5 * wt) x(t) = cos(2t + 1)
Answers
To determine the complex exponential Fourier series representation for each of the given signals, we can use the formula:
X(t) = ∑[n=-∞ to ∞] Cn * e^(jnω0*t)
where Cn represents the complex Fourier coefficients and ω0 is the fundamental frequency.
X(t) = cos(ωt):
The complex exponential Fourier series representation for X(t) = cos(ωt) is:
X(t) = C0 * e^(j0ω0t) + C1 * e^(j1ω0t) + C(-1) * e^(j*(-1)ω0t)
= C0 + C1 * e^(jω0t) + C(-1) * e^(-jω0t)
x(t) = -cos(0.5 * ωt):
The complex exponential Fourier series representation for x(t) = -cos(0.5 * ωt) is:
x(t) = C0 * e^(j0ω0t) + C0 * e^(j0ω0t) + C1 * e^(j1ω0t) + C(-1) * e^(j(-1)ω0t)
= C0 + C0 + C1 * e^(j0.5ω0t) + C(-1) * e^(-j0.5ω0t)
x(t) = cos(2t + 1):
The complex exponential Fourier series representation for x(t) = cos(2t + 1) is:
x(t) = C0 * e^(j0ω0t) + C2 * e^(j2ω0t) + C(-2) * e^(j*(-2)ω0t)
= C0 + C2 * e^(j2ω0t) + C(-2) * e^(-j2ω0t)
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an economy pack of highlighters contains 12 yellow, 6 blue, 4 green, and 3 orange highlighters. an experiment consists of randomly selecting one of the highlighters and recording its color. find the probability that a blue or yellow highlighter is selected given that a yellow highlighter is selected. 0 1
Answers
The probability of choosing a blue or yellow highlighter when a yellow highlighter is picked is 17/25.
Here, according to the question:
Total number of yellow highlighters = 12
Total number of blue highlighters = 6
Total number of green highlighters = 4
Total number of orange highlighters = 3
∴ Total number of highlighters = 12 + 6 + 4 + 3 = 25
Let E1: Event of selecting yellow highlighter.
P(E1) = \(\frac{Total number of yellow highlighters}{Total highlighters}\) = \(\frac{12}{25}\)
Let E2: Event of selecting blue or yellow highlighter one yellow highlighter is selected.
∴Total yellow highlighters left after selecting one = 12 -1 = 11
Also, the number of blue highlighters = 6
So, the TOTAL FAVORABLE to E2 = 6+ 11 = 17
P(E2) = \(\frac{Total number of yellow + blue highlighters}{Total highlighters}\) = \(\frac{17}{25}\)
Hence, The probability of choosing a blue or yellow highlighter when a yellow highlighter is picked is 17/25.
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solve b+4b^2-64a^3 when a= 1/4 and b= 0.2
Answers
Answer:
-1.04.
Step-by-step explanation:
We are asked to simplify the expression with given values.
For this problem, we should use substitution.
What is Substitution?Substitution is replacing an unknown value with a known value.
For example, replacing the variable x with a known value such as 2.
Let's substitute the given values into our expression.
Substitute:
\(-0.2+4(-0.2)^2-64(1/4)^3\)
Evaluate:
\(-0.2+4(0.04)-64(1/64)\)
\(-0.2+0.16-1\)
\(-1.04\)
The solution is -1.04.
Find the missing values a, b, c and d
Answers
Answer:
a answer koStep-by-step explanation:
ur welcome. carry on learningFind the opposite of the opposite of each integer.
5. -4
7. 8
6. -15
8. -7
Answers
Answer:
5: 4
7: -8
6: 15
8: 7
I hope this helps you <3
Step-by-step explanation:
expression is equivalent to
14
−
17
Answers
Answer:
14/17
Step-by-step explanation:
14/17 as a fraction can't be simplified. 17 is a prime number, so it can't be divided by anything except 1.
If the question you're asking is 14-17, the answer would be -3...
I hope this helped you out!! Have an awesome day C:
Step-by-step explanation:
if you are subtracting it is -3
if you are dividing it is 0.823
if you are looking for an equivalent fraction it is 28/34
what is another term for the expected value of a discrete probability distribution
Answers
The expected value of a discrete probability distribution is also known as the mean or the mathematical expectation.
The expected value of a discrete probability distribution is a concept in probability theory that represents the average value that can be expected from a random variable.
It is a measure of the central tendency of the distribution, and represents the average value of the random variable over many trials. The expected value is calculated by summing the product of each possible outcome and its corresponding probability.
It is a weighted average of all the possible outcomes of a discrete probability distribution, where the probabilities of each outcome are used as weights.
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PLEASE HELP
Given f(x)= 6(8-x)+5
(a) what does the notion f(-2) mean?
(b) what is the value of f(-2)?
Answers
b: 65
Write an equation in slope-intercept form with a slope of 7 and a y-intercept of - 3.
Answers
Answer:
y = 7x - 3
Step-by-step explanation:
y = mx + b
m = slope
b = y-intercept
y = 7x - 3
ANSWER ASAP!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
PS: Only answer #8
Answers
Answer:
Sequence 1: Translate 5 units to the right, and 5 units down
Sequence 2: Reflect across the origin, translate 1 unit down, 1 unit to the right
Step-by-step explanation:
Chose 1 point to focus in, such as (-1, 1)
Move that point 5 units to the right, and 5 units down. See how it matches up with the same corner of the red square?
Now, use that same point, and change the signs of both the x and y, this is to reflect across the origin. (In reality, you are just reflecting across the x and y axis)
So you originally have: (-1,1) -> (1, -1)
Now move this point one unit to the right, and 1 unit to the left.
It may seem that you made it touch the wring corner, but do the same thing with another corner's points. They all flipped!
guys help please what is this
Answers
Answer:
A & C
Step-by-step explanation:
You're looking for rational numbers that are also integers, but because (as the figure shows) ALL integers are already rational, you only have to worry about the integer part. An integer is a whole number, positive or negative. So, you need to simplify each element of each set (the numbers inside the curly braces) to see if they're whole.
A - The square root of 100 is just 10 or -10, divided by -2 is -5 or 5. 4, -2, 5, and -5 are all integers.
B - Wrong because 4/3 is not a whole number
C - 16/4 is just 4, and the square root of 49 is 7 or -7. 4, -6, 2, 7, & -7 are all integers.
D - Wrong because 27/2 is not a whole number
E - Wrong because 0.2 is not a whole number
Let me know if you need a more thorough explanation! (:
i need to know the power of 3
Answers
Answer:
The “Cube” or “Exponent 3” or “Power of 3” of a number is the number multiplied by itself 3 times.
Step-by-step explanation:
It is written as number 3. Saying “4 cubed” or “4 to the power of 3” or “4 3 ” is the same as saying 4 x 4 x 4 = 64.
Answer:
729
Step-by-step explanation:
can i have brainliest i need one more to level up
Philip's rectangular deck is 5 yards wide. The ratio of the width to the length is 4 to 5. What is the approximate length of Philip's deck in feet? Hint: 1 yard = 3 feet.
Answers
Answer:
18.75 feet
Step-by-step explanation:
Step 1
we find the length of Philip's rectangular deck in yards
The ratio of the width to the length is 4 to 5.
Philip's rectangular deck is 5 yards wide.
Hence,
Hence:
4/5 = 5/x
Cross Multiply
4x = 5 × 5
4x = 25
x = 25/4
x = 6.25 yards
The length of Philip's field = 6.25 yards
Hint: 1 yard = 3 feet.
Converting the yard to feet
1 yard = 3 feet
6.25 yards = x
Cross Multiply
x = 6.25 × 3 feet
x = 18.75 feet
Therefore, the length of Philip's field in feet = 18.75 feet
I was reading a book last night that said about 2/3 of Americans attend college, and about 40 percent of those actually graduate. So, about what percent of Americans actually earn college degrees?
Answers
Answer:
26.67% Of Americans earn college degrees.
Step-by-step explanation:
\(\frac{2}{3} *40 = 26.67\)
Pls help 8th grade math due at midnight
Answers
Slope=3;Y intercept=5
what do u need to know in order to find velocity
Answers
Answer:
you need to divide the distance by the time it takes to travel that same distance, then you add your direction to it.
Step-by-step explanation:
Simplify completely quantity 4 x squared minus 32 x plus 48 all over quantity 3 x squared minus 17 x minus 6 . (1 point)
Answers
Answer:
answer: the answer is 30 miles per hour.
ft/s -> mphmph = ft/s 1.466667mph = 44 divided by 1.466667mph = 30.00000
r3ktforgood
the correct answer is 5 6 and 7
Step-by-step explanation:
If $500 is deposited into an account that pays 8% and compounded annually. How long will it take for the deposit to double. please show step by step working
Answers
Answer:
4
Step-by-step explanation:
500 times 0.08/8%
It take 12.5 years for the deposit to double.
What is Simple interest?A quick and easy method of calculating the interest charge on a loan is called a Simple interest.
Given that;
$500 is deposited into an account that pays 8% and compounded annually.
Now, Let the time take for the deposit to double = T
So, We can formulate;
We know that;
⇒ Total amount = Principal amount + Simple interest
Here, Principal amount = $500
Total amount = $1,000
⇒ Simple interest = PRT / 100
⇒ Simple interest = 500 × 8 × T / 100
⇒ Simple interest = 40T
Substitute all the values, we get;
⇒ Total amount = Principal amount + Simple interest
⇒ 1,000 = 500 + 40T
⇒ 500 = 40T
⇒ T = 500 / 40
⇒ T = 12.5 years
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What is the inverse of the function f(x) = 2x + 1?
h(x) = one-halfx – one-half
h(x) = one-halfx + one-half
h(x) = one-halfx – 2
h(x) = one-halfx + 2
f(–2) = g(–2)
f(0) = g(–2)
f(–2) = g(0)
Answers
The inverse of the function f(x) = 2x+1 is (x-1)/2.
What is inverse of function?The output of a function is returned by the inverse function, which also returns the initial value. Consider the inverse relationship between the functions f and g: f(g(x)) = g(f(x)) = x. The initial value is fetched via a function that is its inverse.
Given;
Function, f(x) = 2x + 1.
To find the inverse of the function:
First, let y = f(x)
That is,
y = 2x +1
Then, make x the subject of the equation
y = 2x + 1
To make x the subject of this equation:
First, subtract 1 from both sides
We get
y -1 = 2x
Now, divide both sides by 2.
x = (y-1)/2
Now, rewrite as f⁻¹(x) by replacing y by x.
That is,
f⁻¹(x) = (x-1)/2
Therefore, the inverse of the function f(x) is (x-1)/2.
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Evaluate: 5x + 3 for when x = 2
Answers
Answer:
13
Step-by-step explanation:
5 · x + 3 = 5 · 2 + 3 = 10 + 3 = 13