Find the least common denominator for these fractions. Enter your answer in
the space provided.
1/2 and 3/5

The definite integral is [tex]& \frac{243}{5}(\ln 3)^2-\frac{2}{5}\left[\ln 3\left(\frac{243}{5}\right)-\frac{243}{25}\right]-\frac{2}{125}[/tex].

The integral is [tex]$\int_1^3 x^4(\ln x)^2 d x$[/tex]

First evaluate indefinite integral, [tex]$\int x^4(\ln x)^2 d x$[/tex]

The definite integral of a real-valued function f(x) with respect to a real variable x on an interval [a, b] is expressed as [tex]\int_a^b f(x) d x=F(b)-F(a)[/tex]

Here,

∫ = Integration symbol

a = Lower limit

b = Upper limit

f(x) = Integrand

dx = Integrating agent

Thus, ∫ab f(x) dx is read as the definite integral of f(x) with respect to dx from a to b.

Indefinite integrals are implemented when the boundaries of the integrand are not specified.

Use integration by parts:

[tex]$\int u d v=u v-\int v d u$[/tex]

Let [tex]$u=(\ln x)^2$[/tex] and [tex]$a v=x^4 d x$[/tex]

This implies [tex]$d u=\frac{2(\ln x)}{x} d x$[/tex] and [tex]$v=\int x^4 d x=\frac{x^5}{5}$[/tex]

Equation (1) becomes,

[tex]$$\begin{aligned}& \int x^4(\ln x)^2 d x=(\ln x)^2 \frac{x^5}{5}-\int \frac{x^5}{5} \times \frac{2(\ln x)}{x} d x \\& \int x^4(\ln x)^2 d x=(\ln x)^2 \frac{x^5}{5}-\frac{2}{5} \int x^4 \ln x d x\end{aligned}$$[/tex]

Again, use integration by parts to evaluate [tex]$\int x^4 \ln x d x$[/tex]

Let [tex]$u=\ln x, d v=x^4 d x$[/tex]

This implies [tex]$d u=\frac{1}{x}$[/tex] and  [tex]$v=\int x^4 d x=\frac{x^5}{5}$[/tex]

Let [tex]$u=\ln x, d v=x^4 d x$[/tex]

This implies [tex]d u=\frac{1}{x}$[/tex] and [tex]$v=\int x^4 d x=\frac{x^5}{5}$[/tex]

[tex]$$\begin{aligned}\int x^4 \ln x d x & =\ln x\left(\frac{x^5}{5}\right)-\int\left(\frac{x^5}{5}\right) \frac{1}{x} d x \quad \text { since } \int u d v=u v-\int u d u \\& =\ln x\left(\frac{x^5}{5}\right)-\frac{1}{5} \int x^4 d x \\& =\ln x\left(\frac{x^5}{5}\right)-\frac{x^5}{25}\end{aligned}$$[/tex]

From equation (2),

[tex]$$\int x^4(\ln x)^2 d x=(\ln x)^2 \frac{x^5}{5}-\frac{2}{5}\left[\ln x\left(\frac{x^5}{5}\right)-\frac{x^5}{25}\right]$$[/tex]

The required definite integral is,

[tex]$$\begin{aligned}\int_1^3 x^4(\ln x)^2 d x & =\left[(\ln x)^2 \frac{x^5}{5}-\frac{2}{5}\left[\ln x\left(\frac{x^5}{5}\right)-\frac{x^5}{25}\right]\right]^3 \\& =\left[(\ln 3)^2 \frac{3^5}{5}-\frac{2}{5}\left[\ln 3\left(\frac{3^5}{5}\right)-\frac{3^5}{25}\right]\right]-\left[(\ln 1)^2 \frac{1^5}{5}-\frac{2}{5}\left[\ln 1\left(\frac{1^5}{5}\right)-\frac{1^5}{25}\right]\right] \\& =\frac{243}{5}(\ln 3)^2-\frac{2}{5}\left[\ln 3\left(\frac{243}{5}\right)-\frac{243}{25}\right]-\frac{2}{125}\end{aligned}$$[/tex]

Therefore, the required definite integral is [tex]& =\frac{243}{5}(\ln 3)^2-\frac{2}{5}\left[\ln 3\left(\frac{243}{5}\right)-\frac{243}{25}\right]-\frac{2}{125}[/tex].

For more such questions on integrals.

https://brainly.com/question/22008756

#SPJ4

Answer:

Step-by-step explanation:

remember,

x^(a/b) corresponds to

bth root of x^

7.

x^(2/3) × y^(5/3)

= 3rd root of (x²y⁵)

8.

(a³b⁷/(a²b^-3))

= a¹×b¹⁰

so, the whole expression is

a^(1/5) × b^(10/5) = a^(1/5) × b²

9.

(2^(1/2) × 3^(5/2))^(1/3)

= (2^(1/6) × 3^(5/6))

the whole expression is

6th root of (2 × 3⁵)

The term 91 is present in sequence S at n = 43, and the term is not a part of the sequence T.

A sequence is an enumerated group of items in mathematics where repetitions are permitted, and order is important. Similar to a set, it has members.

Given that:

a = 91

Substitute the value of a in the given sequence.

Sequence S:

a = 2n+5

91 = 2n + 5

91 - 5 = 2n

n = 43

Sequence T:

a = 3n - 6

91 + 6 = 3n

n = 97/3

As the terms present in a sequence cannot be a fraction we figure that 91 is not present in sequence T.

Learn more about sequence here:

https://brainly.com/question/21961097

#SPJ1

h = kv is the formula connecting the height reached and the initial velocity

A variation is a relation between a set of values of one variable and a set of values of other variables.

From the information, the stone is thrown upwards, the height, h, it reaches is proportional to the square of the initial velocity, v.

Since the height, h is proportional to the square of the initial velocity, v.

We can write the formula as follows:

h α v

h = kv   (where k is constant of proportionality)

Therefore, the formula connecting the height reached and the initial velocity is h = kv  

Learn more about variation on:

brainly.com/question/2375770

#SPJ1

Answer:$231.2

Step-by-step explanation:

An equation is a mathematical statement that is made up of two expressions connected by an equal sign.

The total cost of farmers' chicks is $231.2

What is an equation?

An equation is a mathematical statement that is made up of two expressions connected by an equal sign.

Example:

2x = 5 is an equation.

We have,

Total chicks = 70

4 Rhode Island Red chicks = 3 Buckeye chicks.

Multiply 10 on both sides.

40 Rhode Island Red chicks = 30 Buckeye chicks.

We see that,

40 + 30 = 70

Now,

Rhode Island Red chicks cost = $3.20

Buckeye chicks cost = $3.44

Now,

= 40 x 3.20 + 30 x 3.44

= 128 + 103.2

= $231.2

Thus,

The total cost of farmers' chicks is $231.2

Learn more about equations here:

brainly.com/question/17194269

#SPJ2

The measure of congruent angles X and Y of the hexagon are 135.

let the measure of the congruent angles of  X and Y be a

Then by the angle sum property of a hexagon sum of all the angle will be

U +V +W +X +Y +Z = 720

then 100 + 110+ 149 + 91 + a + a =720

2a = 720 - 450

2a = 270

Then a = 135

Hence the congruent angles X and Y are 135.

To learn more about congruent from the given link

https://brainly.com/question/24887947

#SPJ1

[tex]26 sin 58°[/tex] is the expression that represents the length of segment PR.

The word "length" is used to describe an object's size or the separation between two points. The length of an object or the separation between two points is measured by its length. It is employed to determine an object's size or the separation between two points.

Since triangle JKL and triangle PQR are similar, we can use the similarity ratios to find the length of PR.

We know that the ratio of PR to JK is the same as the ratio of PQ to JL. We know that [tex]PQ = 26 and JL = 26 + 13 = 39[/tex], so the ratio is [tex]26/39[/tex].

So, [tex]PR = (26/39) * 13 = 26sin( JL/2) = 26sin58°[/tex]

So the answer is [tex]26 sin 58°[/tex]

To know more about triangle visit: https://brainly.com/question/2773823

#SPJ1

Answer: 0.125 Miles

Step-by-step explanation:

X+2.25 +2.125 = 4.5
X+4.375 =4.5
4.5-4.375= X = 0.125

Answer: <C = 32, <D = 80, <DEC = 68

Step-by-step explanation:

First, the exterior angle DEF equals the sum of the two other interior angles that are not adjacent to it, meaning that the angle DEF equals the sum of angles DCE and EDC.

Therefore, you get the equation

2y+8+6y+8=112

8y+16=112

8y = 96

y = 12

Since y is 12, let's use this to find the measure of the other angles.

DCE = 2y+8 = 32

EDC = 6y+8 = 80

Angle DEC equals 180 - 112 = 68

In mathematics, an inequality is a statement that compares two expressions or values.

The expressions or values being compared are separated by an inequality symbol, such as; > (greater than), < (less than), ≥ (greater than or equal to), ≤ (less than or equal to), or ≠ (not equal to).

For example, the inequality statement "3 < 5" means that 3 is less than 5. The symbol "<" is the inequality symbol, and it shows the relationship between the two values 3 and 5. Another example would be "x + 2 > 5", this inequality states that the value of x + 2 is greater than 5, where x can be any value.

Inequality statements can also be represented graphically on a number line. For example, the inequality x > 3 would be represented by a line on the number line with an open circle at 3, with all the numbers to the right of 3 being included in the solution set.

Inequality statements can also be combined to form a system of inequalities, which can then be solved graphically by finding the common solution set of all the inequalities in the system.

Learn more about inequality statements here: brainly.com/question/17712042

#SPJ4

0 is not a prime number. A prime number is defined as a positive integer greater than 1 that has no positive integer divisors other than 1 and itself. If we look at 0 it does not meet this requirement it means it cannot be a prime number.

0 does not meet this definition because it is not greater than 1 and it does have positive integer divisors other than 1 and itself (for example, it is divisible by 2, 4, and so on). Additionally, the concept of prime numbers is defined only for positive integers, 0 is not positive.

Learn more about prime numbers here: https://brainly.com/question/29062994

#SPJ4

The result of the given multiplication of the mixed fraction is found as 78/3.

A number that combines an integer as well as a legal fraction is referred to as a mixed fraction.

The given expression:

= 4 1/3 x 6

Solving the mixed fraction.

= (4*3 + 1)/3 x 6

= (12 + 1)/3 x 6

= 13/3 x 6

= 78/3

Thus, the result of the given multiplication of the mixed fraction is found as 78/3.

To know more about the mixed fractions, here

https://brainly.com/question/28957298

#SPJ1

Answer:

109 it is the answer

Step-by-step explanation:

02947 029377393

The equation of the line which passes through points (2 3) and (-4 1) is 3y - x = 7.

What is the equation of the line?

The formula for a straight line is y=mx+c where c is the height at which the line intersects the y-axis, often known as the y-intercept, and m is the gradient.

Here, we have

Given: points (2 3) and (-4 1)

We have to find the equation of a line.

we need to determine the slope of the line. The slope can be found by using the formula

m = (y₂ - y₁) /(x₂ - x₁)

Now, we put the given points and get the value of m.

m = (1 - 3) /(-4 - 2)

m = 1/3

Now, we can use the point-slope formula to find an equation for the line. The point-slope formula states:

(y - y₁) = m(x - x₁)

(y - 3) = 1/3 (x-2)

3y - 9 = x - 2

3y - x = 7

Hence,  the equation of the line which passes through points (2 3) and (-4 1) is 3y - x = 7.

To learn more about the equation of a line from the given link

https://brainly.com/question/18831322

#SPJ4

equation for a tractrix, hence it proves that the trace of the tractrix is given as (t, t tan t).

The trace of the tractrix is given as:

Trace = (x, y) = (t, t tan t)

To show that the trace is that of a tractrix, we will use the equation for a tractrix, which is given as:

x = t – t tanh t

where t is the angle that the y-axis makes with the vector.

Substituting the value of x in the trace equation, we get:

t – t tanh t = t

Simplifying the equation, we get:

t tanh t = 0

This is the equation for a tractrix, hence it proves that the trace of the tractrix is given as (t, t tan t).

Learn more about equation here

https://brainly.com/question/29657992

#SPJ4

Answer:

The four main types of factoring are the Greatest common factor (GCF), the Grouping method, the difference in two squares, and the sum or difference in cubes.

Step-by-step explanation:

Hope it helps!

The area of triangle whose sides are 9 cm 12 cm and 15 cm is 54 cm².

The formula to be used for calculating of the area of triangle is -

A = ✓s(s-a) (s-b) (s-c), where a, b and c are sides of triangle. S is (a+b+c)/2.

Let us assume side a to be 9 cm, side b to be 12 cm and side c to be 15 cm. Calculating the value of s.

s = 9 + 12 + 15/2

Adding values on Right Hand Side of the equation

S = 36/2

Performing division

S = 18

Keep the values in formula of area -

A = ✓18 × (18 - 9) × (18 - 12) × (18 - 15)

Performing subtraction

A = ✓18 × 9 × 6 × 3

Performing multiplication

A = ✓2916

Taking square root

A = 54 cm²

Thus, the area of triangle is 54 cm².

Learn more about area -

https://brainly.com/question/21735282

#SPJ4

Option C is correct. 1 = x/4 is not an expression. It is an equation. An expression is a combination of numbers, variables, and operators (such as +, -, *, /) that can be simplified or evaluated to a single value, while an equation is a statement that asserts the equality of two expressions.

Expression: In order to indicate the value of something, an expression is defined as a grouping of integers (constant), letters (variables), or their combination connected by operators (+, -, *, /). A mathematical, algebraic, polynomial, or analytical expression can be used.

It does not demonstrate any relationship because it lacks the equals (=) sign. As a result, it lacks left and right sides. An expression can be made simpler by grouping related terms together, or it can be evaluated by replacing the variables with values to get a numerical value.

Equation: Equation refers to a claim of equivalence. In this statement, two expressions are positioned on either side of one another. The value of the relevant variable must be known in order for an equation to be satisfied; this is referred to as the solution or root of the equation.

To know more about difference between expressions and equation visit:

https://brainly.com/question/17499155

#SPJ1

No, it is not possible to draw a triangle with side lengths of 2.5 cm, 4.2 cm, and 8 cm.

The triangle inequality theorem states that for any triangle, the sum of the lengths of the two shorter sides must be greater than the length of the longest side.

In other words, for a triangle with sides of lengths a, b, and c, where c is the longest side, the following must be true: a + b > c, b + c > a, and a + c > b.

In the case of the triangle with side lengths of 2.5 cm, 4.2 cm, and 8 cm, the sum of the lengths of the two shorter sides (2.5 cm + 4.2 cm) is 6.7 cm, which is less than the length of the longest side (8 cm).

Therefore, it is not possible to form a triangle with these side lengths, as the triangle inequality theorem is not satisfied.

To learn more about the inequality theorem:

https://brainly.com/question/1163433

#SPJ4

[tex]x=\cfrac{cos(2y)}{e^y}\hspace{5em}x\left( \frac{\pi }{8} \right)=\cfrac{1}{e^{\frac{\pi }{8}}\sqrt{2}} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{dx}{dy}\implies \stackrel{quotient~rule}{\cfrac{-2sin(2y)e^y-cos(2y)e^y}{(e^y)^2}}\implies \cfrac{-2sin(2y)-cos(2y)}{e^y} \\\\[-0.35em] ~\dotfill\\\\ \left. \cfrac{dx}{dy} \right|_{y=\frac{\pi }{8}}\implies \cfrac{-2sin(\frac{\pi }{4})-cos(\frac{\pi }{4})}{e^y}\implies \cfrac{-3}{e^{\frac{\pi }{8}}\sqrt{2}}\impliedby m[/tex]

now, since we have the value of "x" at π/8 and we know that at that point y = π/8, and we also have the slope at π/8, let's plug all those fellows in the point-slope formula

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\cfrac{\pi }{8}~~ = ~~\cfrac{-3}{e^{\frac{\pi }{8}}\sqrt{2}}\left(x-\cfrac{1}{e^{\frac{\pi }{8}}\sqrt{2}} \right)[/tex]

[tex]y-\cfrac{\pi }{8}~~ = ~~\cfrac{-3}{e^{\frac{\pi }{8}}\sqrt{2}}x+\cfrac{3}{2e^{\frac{\pi }{4}}} \implies \boxed{\cfrac{3}{e^{\frac{\pi }{8}}\sqrt{2}} {\Large \begin{array}{llll} x \end{array}} + {\Large \begin{array}{llll} y \end{array}}-\cfrac{\pi }{8}-\cfrac{3}{2e^{\frac{\pi }{4}}}~~ = ~~0}[/tex]

Answer:

-10/7 or -1.429

Step-by-step explanation:

-5/7 ÷ 2/4

Fractions can't be divided, so you reciprocate and replace the division sign with a multiplication sign.

-5/7 × 4/2

-20/14

= -10/7 or -1.429

The transformation is an example of a translation because the figure slides in one direction, but does not flip, turn, or change size.

Now, According to the question:

Type of Transformation:

There are four types of transformation in geometry, namely translation, reflection, rotation, and dilation. In translation, it slides the figure in any direction while in reflection, it flips the figure over a point or a line. Also, in rotation the figure turns, while in dilation the figure is either enlarged or reduced.

The type of transformation can be defined as moving a figure to a new location, without any change in the figure's size or shape being called translation. In translation, the figure merely slides or moves in the same direction and distance.

Learn more about Transformation at:

https://brainly.com/question/11709244

#SPJ4

The measure of central tendency that would most likely give the best representation for the data would be Median.

The mean (average) would be greatly skewed by the presence of 0's and would not accurately represent the scores of the students who took the test. Since the scores were all above 75, the mean would be much lower than the actual scores due to the presence of 0's.

The mode (most common value) would be 0 and it would not be a good representation of the scores of the students who took the test.

The median can be calculated by ordering the scores and finding the middle value. Since 5 students were absent and their scores were listed as 0, it would be the middle value of the test scores that were above 75. This is therefore the best measure of central tendency.

Find out more on measures of central tendency at https://brainly.com/question/28101345

#SPJ1

The total area of kite LMNP with diagonal length of 6.4 m and 12.4m in square meters is 39.68 square meters.

Therefore the answer is 39.68 square meter.

To find the area of a kite, you need to know the lengths of its diagonals. The formula for finding the area of a kite is

(d1 × d2) / 2

where d1 and d2 are the lengths of the diagonals.

Here diagonals d1 = 6.4m and d2 = 12.4m, so

area = (d1 × d2) / 2

area = (6.4 × 12.4) / 2

area = 39.68 square meters

So area of the kite LMNP is 39.68 square meters.

To know more on area of kites

https://brainly.com/question/15640807

#SPJ4

200

Portions of quantities can be represented as percentages and decimals, which can help solve different types of questions.

Multiplying Decimals

To find any percent of a quantity, you can multiply the quantity by the percentage as a decimal. We can convert percentages to decimals easily. By moving the decimal point 2 places to the left, we can change 40% into a decimal. So, 40% is equal to 0.4. Now, multiply this by 500.

So, Leo has put together 200 pieces.

Multiples of 10

Since 40% is a multiple of 10, we can solve using another method. It is easy to find 10% of any quantity, just move the decimal point 1 place to the left. So, 10% of 500 is 50.

Then, since 40 is 4 times 10, we can say that 40% of 500 is 4 times 10% of 500.

This proves that Leo has done 200 pieces.

The  △ABC and △DEF are congruent by SAS if  BC = EF. Hence option D is the correct option.

What are congruent triangles?

If all three corresponding sides are equal and all three corresponding angles are identical in measure, two triangles are said to be congruent. These triangles can be moved, rotated, flipped, and turned to look exactly the same. They coincide if they are relocated. ' is the symbol of congruence.

Given that in △ABC and △DE:

AB = DE

∠A ≅ ∠D

SAS rule: Two triangles are congruent if two sides and an included angle of one triangle are equivalent to two sides and an included angle of the second triangle.

The opposite side of ∠A is BC and the opposite side ∠D is EF.

Thus the additional information, that is needed is BC = EF.

To learn more about SAS rule, click on the below link:

https://brainly.com/question/5312871

#SPJ1