Simplify the radical √54​

Consider the integral

[tex]\displaystyle \int_{-\infty}^\infty \frac{5 + 2x^2}{1 + x^2} e^{-x^2} \, dx[/tex]

which is the negative of yours. Bit strange to integrate over [tex](\infty,-\infty)[/tex], but if that's what you actually intended, just multiply the final result by -1. Of course, I've already canceled the superfluous factors of [tex]x^2[/tex].

Expand the integrand into partial fractions.

[tex]\displaystyle \int_{-\infty}^\infty \frac{5 + 2x^2}{1 + x^2} e^{-x^2} \, dx = \int_{-\infty}^\infty \left(2 + \frac3{1+x^2}\right) e^{-x^2} \, dx[/tex]

Recall that for [tex]\alpha>0[/tex],

[tex]\displaystyle \int_{-\infty}^\infty e^{-\alpha x^2} \, dx = \sqrt{\frac\pi\alpha}[/tex]

Now let

[tex]\displaystyle I(a) = \int_{-\infty}^\infty \frac{e^{-ax^2}}{1+x^2} \, dx[/tex]

Together, these give

[tex]\displaystyle \int_{-\infty}^\infty \frac{5 + 2x^2}{1 + x^2} e^{-x^2} \, dx = 2\sqrt\pi + 3I(1)[/tex]

Differentiate [tex]I(a)[/tex] under the integral sign with respect to [tex]a[/tex] to obtain a simple linear differential equation.

[tex]\displaystyle \frac{dI}{da} = -\int_{-\infty}^\infty \frac{x^2 e^{-ax^2}}{1+x^2} \, dx \\\\ ~~~~~~~~ = - \int_{-\infty}^\infty \left(1 - \frac1{1+x^2}\right) e^{-ax^2} \, dx \\\\ ~~~~~~~~ = -\sqrt{\frac\pi a} + I(a)[/tex]

Solve for [tex]I(a)[/tex] with the initial value [tex]I(1) = \sqrt\pi[/tex]. Using an integrating factor,

[tex]\displaystyle \frac{dI}{da} - I(a) = -\sqrt{\frac\pi a} \\\\ e^{-a} \frac{dI}{da} - e^{-a} I(a) = -\sqrt{\frac\pi a}\,e^{-a} \\\\ \frac{d}{da}\left[e^{-a} I(a)\right] = -\sqrt{\frac\pi a}\,e^{-a}[/tex]

By the fundamental theorem of calculus,

[tex]\displaystyle e^{-a} I(a) = e^{-a}I(a)\bigg|_{a=0} - \sqrt\pi \int_0^a \frac{e^{-\xi}}{\sqrt\xi} \, d\xi \\\\ I(a) = \pi e^a - \sqrt\pi \, e^a \int_0^a \frac{e^{-\xi}}{\sqrt\xi} \, d\xi[/tex]

so that

[tex]\displaystyle I(1) = \pi e - \sqrt\pi\,e \int_0^1 \frac{e^{-\xi}}{\sqrt\xi} \, d\xi[/tex]

Substitute [tex]t=\sqrt\xi[/tex].

[tex]\displaystyle I(1) = \pi e - 2\sqrt\pi\,e \int_0^1 e^{-t^2} \, dt[/tex]

Recall the error function,

[tex]\mathrm{erf}(x) = \displaystyle \frac2{\sqrt\pi} \int_0^x e^{-t^2} \, dt[/tex]

which we can use to write

[tex]I(1) = \pi e - 2\sqrt\pi e \cdot \dfrac{\sqrt\pi}2\,\mathrm{erf}(1) = \pi e - \pi e \,\mathrm{erf}(1)[/tex]

Finally, we arrive at

[tex]\displaystyle \int_{-\infty}^\infty \frac{5 + 2x^2}{1 + x^2} e^{-x^2} \, dx = \boxed{2\sqrt\pi + 3\pi e - 3\pi e \, \mathrm{erf}(1)}[/tex]

Answer:

length= 12 width=6

Step-by-step explanation:

when adding perimeter you add the length and width twice each so divide 36 by two and get 18

if the length is twice the width then you could make the equation w+2w=18

by adding like terms you get 3w=18

to get n by itself you divide 18 by 3 and get w=6

if the width is 6 and the length is twice that then the length is 12

Answer:

A=72

Step-by-step explanation:

The length of a rectangle is TWICE it's width meaning 2 times. (Note that L= length, W=width, and A=Area)

L=2w

4w+2w=36

6w=36 <- ( 6 is width)

then

2x6=12

Then

12x6=72

So the area of the rectangle is 72.

Hope that makes sense!

Answer:

Yes

Step-by-step explanation:

1 kilogram of soil will fill:

1/3 ÷ 2/5 = 5/6 (of a container)

Because 5/6 < 1

=> 1 kilogram of soil will fit in the container

Answer:

There is no solution.  The lines are parallel.  They will never intercept.  There will be no point that is on both lines.

Step-by-step explanation:

Answer:

5

Step-by-step explanation:

because I give him and the answer is incorrect I know but I gave him believe me

The solution to  function f(x) = 4 is 2 1/5.

Given that, f(x) = 5x - 7

Now, function  f(x) = 4 implies that

i.e. 5x - 7 = 4

i.e. 5x = 4 + 7

i.e. 5x = 11

i.e. x = 11/5

x =2 1/5

So, option (4) is correct. ​​​​​

The core concept of mathematics' calculus is functions. The unique varieties of relations are the functions. In mathematics, a function is represented as a rule that produces a distinct result for each input x. In mathematics, a function is indicated by a mapping or transformation. Typically, these functions are identified by letters like f, g, and h. The collection of all the values that the function may input while it is defined is known as the domain. The entire set of values that the function's output can produce is referred to as the range. The set of values that could be a function's outputs is known as the co-domain.

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The function is f(x)= 5x-7. The solution to f(x)=4 is when x is 11/5.

Given that,

The function is f(x)= 5x-7

A function is nothing but in mathematics, a function is a unique arrangement of the inputs (also referred to as the domain) and their outputs (sometimes referred to as the codomain),where there is exactly one output for every input, and the output can be connected to its input.

f(x), where x is the input, is a common way to refer to a function. A function represented as y = f(x).

We have to find the solution to f(x)=4

Take 5x-7 in the place of f(x) because they are equal.

5x-7=4

5x=4+7

5x=11

x=11/5

Therefore, the solution x value is 11/5.

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Answer:

B.

Step-by-step explanation:

any arithmetic operation with functions just means to do this operation with the functional expressions.

f/g is really just the expression of f divided by the expression of g.

so, this results in a fraction with the denominator (bottom part) = 2x + 1

as you know, a division by 0 is not a valid operation, and therefore we have to forbid any value for x that would create a 0 in the denominator.

so, for what x do we get

2x + 1 = 0

2x = -1

x = -1/2

therefore, we have to exclude x = -1/2.

Answer:

38/39

Step-by-step explanation:

P=s/(s+f)

P=38/(38+1)

P=38/39

Answer:

For the 60-30-90 degree triangle:

x=12

y=[tex]12\sqrt{3}[/tex]

For the 30-60-90 degree triangle:

x=16

y=[tex]8\sqrt{3}[/tex]

For the 45-45-90 degree triangle:

x=7

y=[tex]7\sqrt{2}[/tex]

Step-by-step explanation:

For 30-60-90 triangles, the side opposite the 30 degree angle is x, while the side opposite the hypotenuse is 2x, while the side opposite the 60 degree angle is [tex]x\sqrt{3}[/tex].

For 45-45-90 triangles, the sides opposite the 45 degree angles are x, while the hypotenuse is [tex]x\sqrt{2}[/tex].

The computation shows that the tons of sugar when they'll have same price is 18 tons.

The first company charges $5500 to rent trucks plus an additional fee of $200.25. Thai will be:

= 5500 + (200.25 × s)

= 5500 + 200.25s

where s = tons of sugar

The second company charges $5041 to rent trucks plus an additional fee of $225.75 for each ton of sugar. This will be:

5041 + 225.75s

Then we will equate both of them. This will be:

5500 + 200.25s = 5041 + 225.75s

5500 - 5041 = 225.75d - 200.25s

459 = 25.5s

s = 459/25.5

s = 18

At 18 tons, they'll have same price.

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The Sugar Sweet Company will choose from two companies to transport its sugar to market. The first company charges $5500 to rent trucks plus an additional fee of $200.25 for each ton of sugar. The second company charges $5041 to rent trucks plus an additional fee of $225.75 for each ton of sugar. For want mount if sugar do they charge the same price.

Answer: 34.3%

Step-by-step explanation:

[tex]Given:\ 12/35\ flights\ on\ time[/tex]

To find the percent, divide then multiply by 100:

[tex]\frac{12}{35} = 0.343\times100=\large\boxed{34.3}[/tex]

Answer:

  C.  Multiply by 5, then subtract 7.

Step-by-step explanation:

You want to know the two steps required to solve this 2-step linear equation.

The variable x has 7 added to it, and the sum is divided by 5. To find the value of x, we must undo these operations in reverse order.

To undo division by 5, we must multiply by 5.

Then, to undo the addition of 7, we must subtract 7.

Here is the "work":

  [tex]\dfrac{x+7}{5}=10\qquad\text{given}\\\\x+7 = 50\qquad\text{multiply both sides by 5}\\\\x=43\qquad\text{subtract 7 from both sides}[/tex]

The operations in order of use are ...

  Multiply both sides by 5 first, then subtract 7 from both sides.

__

Additional comment

The properties of equality tell you that you must do the same operation to both sides of the equation.

The total lifetime cost for Miranda to pay off her 4 loans is: $31,337.27.

Interest is the sum of money paid for using someone else's funds. You must pay interest when you borrow money from lenders. You receive interest when you lend money to borrowers. It could be stated in terms of money or the rate of payment. You will learn more about interest in this article, including the different sorts of interest, what they are, and how to determine how much interest will be charged on a loan or a loan amount.

Given that,

At the beginning of each of her four years in college

Miranda took out a new Stafford loan.

Each loan had a principal of $5,500,

An interest rate of 7.5% compounded monthly

A duration of ten years.

Miranda paid off each loan by making constant monthly payments, starting with when she graduated.

All of the loans were subsidized.

The total lifetime cost for Miranda to pay off her 4 loans is: $31,337.27

Therefore, the total lifetime cost for Miranda to pay off her 4 loans is: $31,337.27

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Answer: D 31,337.27

Step-by-step explanation:

Answer:

Hello,

Step-by-step explanation:

[tex]3y^3+20y^2-7y=y(3y^2+20y-7)\\\\=y(3y^2+21y-y-7)\\\\=y(3y(y+7)-(y+7))\\\\=y(y+7)(3y-1)\\[/tex]

Answer:

-1.

Step-by-step explanation:

Rate of change = [ f(2) - f(-1)] / [2 - (-1)]

                          = [(2)^2 - 2(2) + 3)  - ((-1)^2  - 2(-1) + 3] / 3

                           = ( 3 - 6)/3

                            = -1.

Answer:

Step-by-step explanation:

i answered this one already on your other post for it

3 is the constant/slope

5 is the y intercept

The domain is the set of all integers lying in the interval [0 , 200] and range is the set of all reals lying in the interval [800 , 2800].

What is a function?

A function defined from A to B, associates each and every element of the set A tò the element of set B. In mathematics, it is represented using the symbol, f : A -> B. Here, set A is called the domain of the function while set B is called the range of a function.

Calculation of domain and range of a given function

The domain is the set of all possible values of x.  According to this question, the domain of N lies in the interval [0 , 200] i.e. all the integers lying in the given interval. Only integers are chosen because the number of books cannot be in fraction.

The range is the set of possible costs and lies in the interval [800 , 2800] i.e. all the reals lying in the given interval.

Hence, the domain is the set of all integers lying in the interval [0 , 200] and range is the set of all reals lying in the interval [800 , 2800].

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Answer:

the scientific notation of 83300 is 8.33*10^4

Answer:

Step-by-step explanation:

Answer:

C) 2.7 x 10-15

Step-by-step explanation:

Sorry if I'm wrong. hope you're having a great day

The computation shows that the value of (9 x 10−13) − (6.3 x 10−15) is C. 2.7 x 10−15

Simplify (9 x 10−13) − (6.3 x 10−15).

This will be solved as follows. It should be noted that one just have to subtract the values.rhat are given based on the information above.

Therefore, (9 x 10−13) − (6.3 x 10−15).

= 2.7 x 10−15

In conclusion, the correct option is C.

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Answer:

Yes it is a function

Step-by-step explanation:

There is a relationship between x and y. It is not linear but quadratic

Answer:

yes it is a function

Step-by-step explanation:

that is the answer bc I said I was

Step-by-step explanation:

the absolute value function always delivers the positive value of the internal expression, no matter if it is actually negative or already positive.

so,

|-25| = 25

therefore,

-7 + 25 = 18

The area of the shaded triangle will be 11.5 square centimeters.

It deals with the size of geometry, region, and density of the different forms both 2D and 3D.

The area of the square will be given as,

A₁ = 5 x 5

A₁ = 25

The area of the triangles will be given as,

A₂ = 1/2 (1 x 5 + 4 x 3 + 2 x 5)

A₂ = 1/2 (27)

A₂ = 27/2

Then the area of the shaded triangle will be given as,

A = A₁ - A₂

A = 25 - 27/2

A = (50 - 27) / 2

A = 23/2

A = 11.5 square cm

The area of the shaded triangle will be 11.5 square centimeters.

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Answer:

In the graph, excluding the shaded triangle, there are three right angled triangles surrounding the shaded triangle in a square

Area of shaded triangle = area of square - area of three triangles