Use trigonometric substitution to find or evaluate the integral. (Use C for the constant of integration.) x2 - 64 dx . V x + 64 - 8 sec c+(15)+c x
Answers
The evaluated integral is \(32 ln|sec^{(-1)}(x/8) + tan(sec^{(-1)}(x/8))| + C\).
What is integral?
In mathematics, an integral is a fundamental concept in calculus that represents the accumulation or "summing up" of infinitesimally small quantities. It is used to find the total or net value of a continuous function over a given interval or region.
To evaluate the integral \(\int(x^2 - 64) dx\) using trigonometric substitution, we can use the substitution x = 8 sec(θ).
Let's start by finding the derivative of x with respect to θ:
dx/dθ = 8 sec(θ) tan(θ)
Next, we need to express the differential dx in terms of dθ. To do this, we solve for dx:
dx = 8 sec(θ) tan(θ) dθ
Now, substitute these values in the integral:
\(\int(x^2 - 64) dx = \int((8 sec(\theta))^2 - 64)(8 sec(\theta) tan(\theta)) d\theta\\\\= \int(64 sec^2(\theta) - 64)(8 sec(\theta) tan(\theta)) d\theta\\\\= \int(64 sec^3(\theta) tan(\theta) - 64 sec(\theta) tan(\theta)) d\theta\)
Simplifying the integrand:
\(\int(64 sec^3(\theta) tan(\theta) - 64 sec(\theta) tan(\theta)) d\theta\\\\= \int(64 sec(\theta) (sec^2(\theta) tan(\theta) - 1)) d\theta\\\\= \int(64 sec(\theta) (tan^2(\theta) + tan(\theta) - 1)) d\theta\)
We can use the trigonometric identity \(sec^2(\theta) - 1 = tan^2(\theta)\) to further simplify the integrand:
\(\int(64 sec(\theta) (tan^2(\theta) + tan(\theta) - 1)) d\theta\\\\= \int(64 sec(\theta) sec^2(\theta)) d\theta\\\\= 64 \int sec^3(\theta) d\theta\)
Now, we can evaluate this integral using the trigonometric identity:
\(\int sec^3(\theta) d\theta = (1/2) ln|sec(\theta) + tan(\theta)| + C\)
Substituting back \(\theta = sec^{(-1)}(x/8):\)
\(\int (x^2 - 64) dx = 64 ∫sec^3(\theta) d\theta = 64 (1/2) ln|sec(\theta) + tan(\theta)| + C\)
Replacing θ with \(sec^{(-1)}(x/8):\)
\(= 32 ln|sec(sec^{(-1)}(x/8)) + tan(sec^{(-1)}(x/8))| + C\\\\= 32 ln|sec^{(-1)}(x/8) + tan(sec^{(-1)}(x/8))| + C\)
Thus, the evaluated integral is \(32 ln|sec^{(-1)}(x/8) + tan(sec^{(-1)}(x/8))| + C.\)
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Related Questions
15. What is missing in the problem, 48 is ____ % of 120? A. base B. percentage C. rateD. time
Answers
The following equation-definition will be consider to solve the problem:
\(Base*Rate=Part\)Where the base in this case is 120.
The rate is the porcentage, which is the missing part in the problem.
And 48 is the part. Hence, the solution is:
The solution is C. rate
In a group of 60 tudent, 25 play table tenni, 16 do wimming and 22 play cricket, 8 play table tenni and do wimming, 6 play cricket and do wimming, 5 play table tenni and cricket, and 12 tudent do not play any of thee game
Answers
12 students do not play any of the games. Number of students who do not play any of the games = 60 - 82 = 12
A group of 60 students, 25 play table tennis, 16 do swimming and 22 play cricket, 8 play table tennis and do swimming, 6 play cricket and do swimming, 5 play table tennis and cricket.
25 play table tennis
16 do swimming
22 play cricket
8 play table tennis and do swimming
6 play cricket and do swimming
5 play table tennis and cricket
Find the number of students who do not play any of the games
Total number of students = 60
Number of students who play any of the games = 25 + 16 + 22 + 8 + 6 + 5 = 82
Number of students who do not play any of the games = 60 - 82 = 12
12 students do not play any of the games.
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Find f. Use C for the constant of the first anti derivative and d for the constant of the second derivative
Answers
we have the second derivative of the function f(x)
\(f^{\doubleprime}(x)=2x+9e^x\)step 1
Find out the first antiderivative of the given function
\(f^{\prime}(x)=\int 2x+9e^x=x^2+9e^x+C\)step 2
Find out the second antiderivative
\(f(x)=\int x^2+9e^x+C=\frac{x^3}{3}+9e^x+Cx+D\)therefore
\(f(x)=\frac{x^3}{3}+9e^x+Cx+D\)convert the angle =100 degrees to radians
Answers
Answer:
1.75 rad
Step-by-step explanation:
° × π/180 = rad
100° × π=314.16
314.16/180=1.75 rad
Express the inequality −1.8≤x≤−0.03 using interval notation.
Answers
Answer:
[-1.8,0.03]
Step-by-step explanation:
This is how you use interval notation, square brackets means including the numbers, circular brackets means not including
Answer:
(-1.3,1.8]
Step-by-step explanation:
The inequality −1.3<x≤1.8 means "all numbers between −1.3 and 1.8." In interval notation, we express this as (−1.3,1.8]. Notice that we use a parenthesis to show that the number −1.3 is not included and a bracket to show that the number 1.8 is included.
the area of one lateral face of a right pyramid with an equilateral triangular base is 75 square meters. if the slant height is 30 meters, what is the length of the side of its base, in meters?
Answers
A right pyramid with an equilateral triangular base has a lateral face with a surface area of 75 square meters. The side of the base of a slant with a height of 30 meters is 5 meters long.
Let’s call the side length of the equilateral triangular base “s”. We know that the area of one lateral face is 75 square meters, and that face is a triangle with base s and height 30 meters (the slant height). We can use the formula for the area of a triangle to solve for s:
Area = (1/2) * base * height
75 = (1/2) * s * 30
Simplifying the equation, we get:
75 = 15s
s = 75/15
s = 5
Therefore, the length of the side of the equilateral triangular base is 5 meters.
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Triangle ABC has coordinates A(3, 2), B(5, 5), and C(4, 1). If the triangle is reflected over y = x, what are the coordinates of C'?
Answers
Answer: (1, 4)
Step-by-step explanation:
When you reflect over y=x, (x, y) turns into (y, x).
So the coordinates of C' are (1, 4).
Solve the absolute value equation. |x-8| +7=9
Answers
Hi there! :)
\(\large\boxed{x = 6, 10}\)
|x - 8| + 7 = 9
Subtract 7 from both sides:
|x - 8| = 2
Solve for both solutions by evaluating the expression as negative and positive:
x - 8 = 2
x = 10
-(x - 8) = 2
Divide both sides by -1:
x - 8 = -2
x = 6.
Therefore, the solutions are:
x = 6, 10
If you can help me, I would be very happy.
Answers
Answer: 116 round that to the nearest hundreth.
I hope this helped you well.
A bank account earns interest at a rate of 3.5% per year (in other words it increases in value by that percent) and starts with a balance of $350. Which of the following equations would give the account’s worth, W, as a function of the number of years, y, it has been gaining interest?
Answers
Answer:
Annually cumulating interest can be determined by the following formula:
W=P(1+r)^y
r represents the interest rate as a decimal, and P represents the starting amount of money.
Step-by-step explanation:
Order the set of numbers from least to greatest.
-0.62
-0.615
-5/6
Answers
Answer:
5/6, -0.62, -0.615
Step-by-step explanation:
Hope this Help If i got it wrong I'm sorry I tried :(
In a simple random sample of size 64, there were 38 Individuals in the category of interest. Compute the sample proportion p. Round the answer to at least three decimal places The sample proportion is
Answers
the sample proportion is approximately 0.594.To compute the sample proportion, we need to divide the number of individuals in the category of interest by the total sample size.
In this case, population there were 38 individuals in the category of interest, and the sample size was 64. Therefore, the sample proportion is:
p = 38/64
p = 0.59375
Rounding to three decimal places, we get:
p ≈ 0.594
So, the sample proportion is approximately 0.594.
The sample proportion is an estimate of the population proportion, which tells us the proportion of individuals in the entire population that belong to the category of interest. The larger the sample size, the more accurate the estimate is likely to be. We can use the sample proportion to construct a confidence interval for the population proportion or to test hypotheses about the population proportion.
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-2.3c-6.6=-12.2-3. solve for me please ( show work )
Answers
Answer:
c = 3.73913043
Step-by-step explanation:
Subtract 3 from - 12.2
Move all terms not containing c to the right side of the equation.
− 2.3 c = − 8.6
Divide each term in − 2.3 c = − 8.6 by − 2.3 and simplify.
Solve.
(4 x 104)(2 x 103)
Answers
Answer:
85,696
Step-by-step explanation:
4 x 104 = 416
2 x 103 = 203
No sign between parentheses means that you have to multiply the two outcomes of the parentheses.
416 x 203 = 85,696
f(x,y)=−4x2+4xy−2y2+6x+8y Suppose that there is a constraint such that xy≤15, What is the value of y that optimizes the function? Question 17 f(x,y)=−4x2+4xy−2y2+6x+8y Suppose that there is a constraint such that xy≤15, what is the Lagrange Multiplier?
Answers
The value of y that optimizes the function is y = 3/2. The Lagrange Multiplier in this case is -105/8.
To optimize the function \(f(x, y) = -4x^2 + 4xy - 2y^2 + 6x + 8y\) subject to the constraint xy ≤ 15, we can use the method of Lagrange multipliers.
The Lagrangian function is defined as:
L(x, y, λ) = f(x, y) - λ(g(x, y) - 15)
where g(x, y) = xy is the constraint equation.
To find the optimal value of y, we need to solve the following system of equations:
∂L/∂x = -8x + 4y + 6 - λy = 0 (1)
∂L/∂y = 4x - 4y + 8 - λx = 0 (2)
g(x, y) - 15 = xy - 15 = 0 (3)
From equation (1), we can rearrange it as:
-8x + 4y - λy = -6 (4)
From equation (2), we can rearrange it as:
4x - 4y - λx = -8 (5)
To solve this system of equations, we can eliminate λ by multiplying equation (4) by x and equation (5) by y, and then subtracting the resulting equations:
\(-8x^2 + 4xy - λxy = -6x (6)\\4xy - 4y^2 - λxy = -8y (7)\)
Subtracting equation (7) from equation (6), we get:
\(-8x^2 + 4xy - 4y^2 = -6x + 8y\)
Simplifying this equation gives:
\(-8x^2 - 4y^2 + 4xy = -6x + 8y\)
Rearranging terms, we have:
\(-8x^2 + 4xy + 6x - 4y^2 - 8y = 0\)
Factoring, we get:
(2x - y)(-4x - 2y + 6) = 0
Setting each factor equal to zero gives:
2x - y = 0 (8)
-4x - 2y + 6 = 0 (9)
From equation (8), we can solve for y:
y = 2x (10)
Substituting equation (10) into equation (9), we get:
-4x - 4x + 6 = 0
Simplifying, we have:
-8x + 6 = 0
Solving for x, we find:
x = 3/4
Substituting this value of x into equation (10), we can find y:
y = 2 * (3/4) = 3/2
Therefore, the value of y that optimizes the function is y = 3/2.
To find the Lagrange Multiplier, we substitute the optimal values of x and y into the constraint equation (3):
xy - 15 = (3/4) * (3/2) - 15 = 9/8 - 15 = -105/8
Hence, the Lagrange Multiplier is -105/8.
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x + 7y = 6x-(2/3)
Solve for x and y. You do not need to show your work
Answers
Answer:
\( \displaystyle x = \frac{21y + 2}{15} \)
\( \displaystyle y = \frac{15x - 2}{21} \)
Step-by-step explanation:
Solve for x:remove parentheses:
\( \displaystyle x + 7y = 6x - \frac{2}{3} \)
move x to left hand side and y to right hand side and change its sign:
\( \displaystyle x - 6x = -7y - \frac{2}{3} \)
simplify Substraction:
\( \displaystyle - 5x = \frac{-21y - 2}{3} \)
divide both sides by -5:
\( \displaystyle x = \frac{21y + 2}{15} \)
Solve for y:move x to the right hand side and change its sign:
\( \displaystyle 7y = 6x - x- \frac{2}{3} \)
simplify Substraction:
\( \displaystyle 7y =5x- \frac{2}{3} \)
simplify substraction:
\( \displaystyle 7y = \frac{15x - 2}{3} \)
divide both sides by 7:
\( \displaystyle y = \frac{15x - 2}{21} \)
and we are done!
asap! please help i’ll give brainliest
Answers
Answer:
the last one
Step-by-step explanation:
because the line inside the box of the box and whisker plot determines the median!!!
hope this helps!! :D
Answer:
80444444444444444448----2-19999999999999999999999999999999999999488888888888888888888888865194974037205702795042-73--457-309703407203570-2-5-593759597207092-7503928077475073703594370975909304575407590565-17596332-659549579-27943595396575-2759650-2650-2959-92552820595907979474237947479023333333372222227403794322225409732222224700007945479470000374444444440944453709433337094444444444444474555555709479000447950497439270007454709490375777772093749930579047093479352974300000040357999994794432099999994907777743799949034970349743972437900000000940733320944444755555554099970304937777449037709437453094490377974374903
Step-by-step explanation:
Electricty what is the voltage V of an electron circuit with a current C of 2-j and an impedence i of 3+2j? Use the formula v=ci
Answers
The value of the voltage of the electricity is 6 + j + 2j²
How to determine the voltage of the electricity?From the question, we have the following parameters that can be used in our computation:
C = 2 - j
I = 3 + 2j
The formula of voltage is given as
V = CI
Substitute the known values in the above equation, so, we have the following representation
V = (2 - j) * (3 + 2j)
Open the brackets
So, we have
V = 6 + 4j - 3j + 2j²
Evaluate the like terms
V = 6 + j + 2j²
Hence, the voltage is 6 + j + 2j²
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6(2-9)=18-n
...................
Answers
Answer:
10=8n
Step-by-step explanation:
Answer:
n=60
Step-by-step explanation:
(08.02)Using the following data, calculate the mean absolute deviation:
10, 7, 1, 6, 7, 9, 8, 10, 2, 5
What is the mean absolute deviation for these data?
Answers
Answer:
7
Step-by-step explanation:
10+7+1+6+7+9+8+10+2+5
-----------------------------------
10
65 ÷ 10
= 6.5, round up, = 7
Answer:
I took the quiz and the answer is 7
Step-by-step explanation:
How do you decide which technique to use when solving an equation?
Answers
Completing the square – can be used to solve any quadratic equation. It is a very important method for rewriting a quadratic function in vertex form. Quadratic formula – is the method that is used most often for solving a quadratic equation.
What is equation?In its most basic form, an equation is a mathematical statement that indicates that two mathematical expressions are equal. 3x + 5 = 14, for example, is an equation in which 3x + 5 and 14 are two expressions separated by a 'equal' sign. A mathematical statement made up of two expressions joined by an equal sign is known as an equation. 3x - 5 = 16 is an example of an equation. We get the value of the variable x as x = 7 after solving this equation.
Here,
Any quadratic problem may be solved by completing the square. It is a critical way for expressing a quadratic function in vertex form. The quadratic formula is the most often used method for solving a quadratic problem.
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Hugo works in a shop. His normal rate of pay is $9.00 per hour. When Hugo works on Sundays, he is paid overtime for each hour. The overtime rate of pay is 1; times his normal hourly rate. Last weekend, Hugo worked 7 hours on Saturday and 4 hours on Sunday. Work out how money Hugo earned during that weekend.
Answers
Answer: $130
Step-by-step explanation:
Answer:
£111
Step-by-step explanation:
i tried it, its right
A taxi charges $4 plus $0. 20 per mile. You spend more than $30 for a taxi ride. What is the inequality?
Answers
Answer:
\(4+0.20m > 30\)
Step-by-step explanation:
You add 4 to the 2 cents
Sunflower produce approximately 50 seeds per flower. If one ounce of sunflower seeds contain an average of 72 seeds, how many flowers are needed to produce 2 pounds of seeds?
Answers
Given :
Sunflower produce approximately 50 seeds per flower.
If one ounce of sunflower seeds contain an average of 72 seeds.
To Find :
How many flowers are needed to produce 2 pounds of seeds.
Solution :
1 pound = 16 ounces .
So , 2 pound = 32 ounces .
Number of seeds in 2 pounds of seeds, n = 32×72 = 2304 .
Number of flowers are :
\(N=\dfrac{2304}{50}=46.08\)
So , approximate number of flower required are 46.
Hence, this is the required solution.
Determine if the two lines y = 2x - 3/2 and y = -2x + 2/3 are parallel, perpendicular or neither. Explain
Answers
Answer:
Neither
Step-by-step explanation:
For two lines to be parallel, they must have the same slope. 2 and -2 are not the same. For lines to be perpendicular, their slopes must be negative reciprocals (ex 5 and -1/5) The slopes of these lines are not the same or negative reciprocals, so they are not parallel or perpendicular.
Hope this helps :-)
In about months, the populations of the two towns will be equal.
Answers
Help!! Plz complete these two :)
Answers
Determine the type of triangle that is drawn below
Answers
Step-by-step explanation:
The triangle is an Isosceles Right.
researcher records the following scores for an Olympic gymnast following her routine: 9.9, 9.8, 9.6, 9.5, 9.7, 9.1, 8.9, and 9.8. What is the range for the scores?
1.0 (9.9 to 8.9)
0.3 (9.8 to 9.5)
0.5 (9.6 to 9.1)
It is not possible to compute a range with an even number of scores.
Answers
The range for the scores is 1.0 (from 9.9 to 8.9). The range is the difference between the highest and lowest numbers in a set of numbers. In this case, the highest score is 9.9 and the lowest score is 8.9, so the range is 1.0.
In mathematics, the range of a function can refer to one of two similar terms:
the common area of the function
The image of the function
Given two groups X and Y, the binary relation f between X and Y is a (exact) function (X to Y), if there is a y in Y for every x in X, so f is associated with y. The sets X and Y are called the area of f and the common domain, respectively.
The range is a measure of dispersion in a set of numbers. To find the range, you need to subtract the lowest score from the highest score. In this case, the scores for the Olympic gymnast are: 9.9, 9.8, 9.6, 9.5, 9.7, 9.1, 8.9, and 9.8.
First, identify the highest and lowest scores:
Highest score: 9.9
Lowest score: 8.9
Next, subtract the lowest score from the highest score:
Range = 9.9 - 8.9
The range for the scores is 1.0 (9.9 to 8.9).
Your answer: 1.0 (9.9 to 8.9)
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IUITUI Rauuild Ivoiribers Select the number line that shows that two opposite numbers have a sum of 0. 5 O A. -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 2 OB. + 8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 6 11 C. -8 -7 -6.-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 O D. PREVIOUS
Answers
It's important to know that the sum of two opposite numbers is zero because opposite numbers have the same absolute value but opposite sign, for example, 2 and -2 are opposite numbers.
The correct number line must have its final position on zero.
Notice that the number line A shows -5 as the initial number and then goes back 5 units to zero.
Therefore, the answer is A.