Mathematics High School

prove the formula where m and n are positive integers
-x∫x sin mx sin nx dx = 01 m ≠ n
π1 m=n

Answers

Answer 1

The formula that you are asking about is proven using integration by parts.

How to prove the formula

This can be done by setting u = sin(mx) and dv = sin(nx)dx.

Then du = mcos(mx)dx and v = -(1/n)cos(nx).

The integral becomes -xsin(mx)cos(nx) + (1/n)∫xcos(mx)cos(nx)dx.

The next step is to use trigonometric identities to simplify the integral.

This can be done by noting that cos(m+n)x = cos(mx)cos(nx) - sin(mx)sin(nx). Thus, ∫xcos(mx)cos(nx)dx = (1/(m+n))∫xcos(m+n)x dx.

Substituting this back into the original integral and simplifying, we get the formula you provided:

x∫x sin mx sin nx dx = 01 m ≠ n π1 m=n.

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Related Questions

The highest weekly wage of a group
of employees is $$$105.40. If the
range of the wages is $27.50, how
much does the lowest paid employee
receive?
(A.) $105.40
(B.) $77.90
(C.) $66.40
(D.) $27.50

Answers

Range: lowest number subtracted from highest number
105.40 - 27.50 = $77.90

The range is the highest subtracted by the lowest so the answer would be 77.90

Ellie drove 416 miles on 18.7 gallons of gas. What was her gas mileage, rounded tothe nearest mpg?Your Answer:

Answers

Answer:

$22\text{ mpg}$

Step-by-step explanation:

To determine what was her gas mileage, divide the total miles driven by the gallons of gas;

$\frac{416}{18.7}=22\text{mpg}$

A wireless store owner takes presale orders for a new smartphone and tablet. He gets 370 preorders for the smartphone and 210 orders for the tablet. The combined sale of the preorders is $364,410. If the price of a smartphone and a tablet together is $1413, how much does each device cost

Answers

Each devices cost $423 and $990 based on the pre-orders cost and complete order.

Let us assume the smartphone and tablet cost be x and y. The total price of smartphone and tablet is -

x + y = 1413

y = 1413 - x

The total cost on pre-orders is -

370x + 210y = 3,64,410

Keep the value of y from previous equation

370x + 210 (1413 - x) = 3,64,410

Multiply the bracket with 210 on Left Hand Side of the equation

370x + 2,96,730 - 210x = 3,64,410

Rearranging the equation and performing subtraction

160x = 3,64,410 - 2,96,730

160x = 67,680

Divide the values on Right Hand Side

x = 423

Keep the value of x in original equation to find y

y = 1413 - 423

Subtract the values

y = 990

Hence, smartphone will cost $423 and tablet will cost $990.

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Which expression is equivalent to 6 + 12x?

A
3(4x + 1)

B
4 + 3x + x + 8x + 2

C
6(1 + x)

D
4x + 8 + x + x + 4

Answers

B is the correct answer because you add 3x plus x plus 8x to get 12x and then add the 4 and 2 together to get 6, so you end up with 6+12x!!

393/9 with remainder

Answers

Answer:

43 remainder of 6.

Step-by-step explanation:

0.5 is 2.5% of what number?

Answers

Answer:

The number is : 20

Step-by-step explanation:

Thanks alot

Answer:

the number is = 20

Step-by-step explanation:

good bye

yanni paints a room that has 400 square feet of wall space in 2 hours. at this rate, how long will it take her to paint a room that has 720 square feet of wall space?

Answers

Answer: 3 hours and 36 minutes

Step-by-step explanation:

720/400 = 1.8

1.8*2 = 3.6

3.6 hours = 3 hours and 36 minutes

Suppose if you have a lot of training data from an arbitrary distribution, would you expect the lda classifier to give similar boundaries to the bayes classifier?

Answers

No, the LDA classifier and the Bayes classifier would not necessarily give similar boundaries when trained on a large amount of training data from an arbitrary distribution.

The LDA (Linear Discriminant Analysis) classifier assumes that the input features are normally distributed and that the class covariances are equal. It finds linear boundaries that maximize the separation between classes based on these assumptions. On the other hand, the Bayes classifier makes decisions based on the class conditional probabilities and prior probabilities of the classes. It can handle arbitrary distributions and does not make assumptions about the class covariances. Therefore, even with a large amount of training data, if the distribution of the data does not conform to the assumptions of LDA (e.g., non-normal distributions or unequal class covariances), the LDA classifier may not give similar boundaries to the Bayes classifier.

The LDA and Bayes classifiers may not give similar boundaries when trained on data from an arbitrary distribution, as the LDA classifier relies on specific assumptions about the data distribution that may not hold true in practice.

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What is the positive square root of 16?

Answers

Answer:

Step-by-step explanation:

because The square root of 16 is 4, which is a rational number.

Answer:

Step-by-step explanation:

NEED BY MONDAY PLEASE!!! Select all the true equations:

Answers

#2 is correct because cos = adjacent over hypotenuse which in this case is y/15

#3 is correct, because tangent is equal to opposite over adjacent, which in this case is y/x

#4 is correct because sign is opposite over hypotenuse which in that case is X over 15

#5 is correct because the tangent of 63 degrees is equal to opposite over adjacent which is X over Y

The scale factor of two similar polygons is given. Find the ratio of their perimeters and the ratios of their areas.

1) 3:1
2) 7/4

Answers

The ratios of their perimeters and the ratios of their areas are 1) 3:1 and 9:1 and 2) 7:4 and 49:16.

Given the scale factor of two similar polygons, we need to find the ratio of their perimeters and the ratios of their areas,

To find the ratio of the perimeters of two similar polygons, we can simply write the scale factor as it is because the ratio of the perimeter is equal to the ration of the corresponding lengths.

1) So, perimeter = 3:1

The ratio of areas between two similar polygons is equal to the square of the scale factor.

Since the scale factor is 3:1, the ratio of their areas is:

(Ratio of areas) = (Scale factor)² = 9/1 = 9:1

Similarly,

2) Perimeter = 7:4

Area = 49/16

Hence the ratios of their perimeters and the ratios of their areas are 1) 3:1 and 9:1 and 2) 7:4 and 49:16.

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203/4108 as a percent
a. 4.94%
b. 20.24%
c. 0.049%
d. 49.42%

Answers

4.94% is the correct one.

Consider triangle QRS. The legs each have a length of 10 units. Triangle Q R S is shown. Angle Q S R is 90 degrees. The length of Q X is 10 and the length of S R is 10. What is the length of the hypotenuse of the triangle?

Answers

Answer:

10 square root 2

Step-by-step explanation:

Answer: D). 10 StartRoot 2 EndRoot units

Step-by-step explanation:

there are 10 students in the class. two th them should be selected as the best students in the class. how many different ways can be done

Answers

There are 45 different ways in which two of them can be selected as the best students in the class.

A combination in mathematics is a choice made from a group of separate elements where the order of the selection is irrelevant (unlike permutations).

A k-combination of a set S is officially defined as a subset of S's k unique elements.

Total number of students = 10

Number of students to be selected = 2

No. of ways = 10C2

= (10*9)/2

= 5*9

= 45

Hence, there are 45 different ways in which two of them can be selected as the best students in the class.

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5(x + 4) + 3(x + 2) Expanded and simplified

Answers

8x + 26

5(x+4) + 3(x + 2)

5x + 5 x 4 + 3x + 3 x 2

5x + 20 + 3x + 6

* 8x + 6 *

Answer:

$\mathrm{8x+26}$

Step-by-step explanation:

$\mathrm{5(x + 4) + 3(x + 2)}$

Expand:-

$\mathrm{5x+20+3x+6}$

Combine like terms:-

$\mathrm{5x+3x+20+6}$

$\mathrm{(5x+3x)=\bf 8x}$

$\mathrm{(20+6)=\bf 26}$

$\mathrm{8x+26}$

______________________

Hope this helps!

PLEASE HELLPP I WILL MARK BRAINLYLIST

Answers

Answer:

Step-by-step explanation:

divide one side of the bigger triangle by the corresponding side of the smaller one

Answer:

Step-by-step explanation:

Divide the bigger triangle by the little one

3/1=3

then 12/4=3

Put the following numbers in order from greatest to least. -3, -6, 2, 0, -1 4/5

Answers

The first step to solve this exercise is to write the Mixed number as a decimal number. You can do it by dividing the numerator by the denominator of the fractional part and adding the quotient to the Whole number part, as follows:

$1+\frac{4}{5}=1+\text{0}.8=1.8$

So:

$-1\frac{4}{5}=-1.8$

Knowing that Positive numbers are greater than zero and knowing that Negatives numbers are less than zero, we can order the given numbers from greatest to least, as you can see below:

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Honestly I don't know what to do at this point

You are trying to find the distance d across the river, but your map is incomplete. ABC is similar to EDC

Answers

Answer:

175 ft

Step-by-step explanation:

So because these triangles are similar, you can use ratios to find the distances. For the smaller triangle, 100/700 is a 1 to 7 ratio. From that, because it's the longer side, you can divide 1225 by 7 to find d, which is 175 ft.

Answer:

175 ft

Step-by-step explanation:

2. Consider the data points (−1,−1),(2,2) and (1,0). (a) Find the least squares line of best fit for the data. (b) Compute the least-squares error.

Answers

(a) the equation of the least squares line is y = (3/5)x - 1/5. b) Hence, the least-squares error for the given data points is 26/5.

To find the slope, we use the formula:

m = (nΣxy - ΣxΣy) / (nΣx^2 - (Σx)^2)

Using the given data points:

Σx = -1 + 2 + 1 = 2

Σy = -1 + 2 + 0 = 1

Σxy = (-1)(-1) + (2)(2) + (1)(0) = 5

Σx^2 = (-1)^2 + 2^2 + 1^2 = 6

n = 3

Substituting these values into the formula, we get:

m = (3(5) - (2)(1)) / (3(6) - (2)^2) = 3/5

To find the y-intercept, we use the formula:

b = (Σy - mΣx) / n

Substituting the values, we get:

b = (1 - (3/5)(2)) / 3 = -1/5

Therefore, the equation of the least squares line is y = (3/5)x - 1/5.

(b) To compute the least-squares error, we need to calculate the sum of the squared differences between the observed y-values and the predicted y-values using the least squares line.

Using the given data points and the equation of the least squares line, we calculate the squared differences and sum them up:

(-1 - (-1/5))^2 + (2 - (6/5))^2 + (0 - (2/5))^2 = 26/5

Hence, the least-squares error for the given data points is 26/5.

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What is the value of the expression x² + y² given x = 10 and y = 2?

A. 104
B. -98
C. 89
D. -89

Answers

It’s (a) 104

You’ll just replace the letters for the values given (10 and 2) next solve the exponents add your results together and you to 104

Answer:

A. 104

Step-by-step explanation:

Solve for x.

Your answer must be simplified.

-3

Answers

You have to simplify-3?

Answer:

x > 7

Step-by-step explanation:

-3 < x - 10

Subtract x from both sides.

-x - 3 < -10

Add 3 to both sides.

-x < -7

Divide both sides by -1. Remember that when you multiply or divide both sides of an inequality by a negative number, the inequality sign changes direction.

x > 7

Martha earns $17 an hour. How many dollars will Martha earn after 25 hours?

Answers

Martha will earn $425 after 25 hours of work at a rate of $17 per hour.

How many dollars will Martha earn after 25 hours?

Given that, Martha earns $17 an hour.

To determine how much Martha will earn after 25 hours of work, we can use the formula:

Earnings = rate × time

Where "rate" is the hourly rate of pay and "time" is the number of hours worked.

Substituting the given values, we get:

Earnings = rate × time

Earnings = $17/hour × 25 hours

Earnings = $17 × 25

earnings = $425

Therefore, Martha will earn $425 after 25 hours.

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Can y’all help me answer these questions

Answers

I’m sorry what is it that you want me to answer for you I don’t understand? the images won’t load?

Help me please...
Mary took 7 hours to read 210 pages
the relationship between x, the number of hours she reads, and y,the number of pages read was 1x=30y.
How long will it take Mary to read 80 pages at the same reading rate? Show or explain how you got your answer.

Answers

Answer:

2 2/3

Step-by-step explanation:

80/30=8/3=2 2/3

EASY TRIG MATHEMATICS QUESTION! PLEASE HELP ME SOLVE! THANK YOU!

LOOK AT IMAGE FOR DETAILS !

Answers

Answer: YES

Step-by-step explanation:

Polygon ABCD is defined by the points A(-4, 2), B(-2, 4), C(1, 3), and D(2, 2). Match the coordinates of the points of the transformed polygons to their correct values. the coordinates of D′ if polygon ABCD rotates 90° counterclockwise to create A′B′C′D′ (-2, 2) the coordinates of C″ if polygon ABCD rotates 90° clockwise to create A″B″C″D″ (4, -2) the coordinates of A′′′ if polygon ABCD rotates 180° clockwise to create A′′′B′′′C′′′D′′′ (3, -1) the coordinates of B″ if polygon ABCD rotates 270° counterclockwise to create A″B″C″D″ (4, 2) arrowBoth arrowBoth arrowBoth arrowBoth

Answers

Answer:

The answer is below

Step-by-step explanation:

Transformation is the movement of a point from its initial location to a new location. Types of transformation are rotation, reflection, translation and dilation.

If a point A(x, y) is rotated 90° counterclockwise, the new point is at A'(-y, x).

If a point A(x, y) is rotated 90° clockwise, the new point is at A'(y, -x).

If a point A(x, y) is rotated 180° counterclockwise, the new point is at A'(-x, -y).

If a point A(x, y) is rotated 270° counterclockwise, the new point is at A'(y, -x).

Polygon ABCD is defined by the points A(-4, 2), B(-2, 4), C(1, 3), and D(2, 2).

The coordinates of D′ if polygon ABCD rotates 90° counterclockwise to create A′B′C′D′ is D'(-2, 2)

The coordinates of C″ if polygon ABCD rotates 90° clockwise to create A″B″C″D″ is C"(3, -1).

the coordinates of A′′′ if polygon ABCD rotates 180° clockwise to create A′′′B′′′C′′′D′′′ is A''"(4, -2)

the coordinates of B″ if polygon ABCD rotates 270° counterclockwise to create A″B″C″D″ is B"(4, 2)