Help picture below problem 15

The first integral has a well-known beta function representation, so the second one should too. The beta function itself is defined as

[tex]B(x,y) = \displaystyle \int_0^1 t^{x-1} (1-t)^{y-1} \, dt[/tex]

and satisfies the identity

[tex]\displaystyle B(x,y) = \frac{\Gamma(x) \Gamma(y)}{\Gamma(x+y)}[/tex]

Later on, we'll also use the so-called reflection formula for the gamma function; for non-integer z,

[tex]\Gamma(z) \Gamma(1-z) = \dfrac{\pi}{\sin(\pi z)}[/tex]

as well as the identity

[tex]\dfrac{\Gamma(z+1)}{\Gamma(z)} = z[/tex]

Replace [tex]x\to\sin^{-1}(x)[/tex] in both integrals, so that

[tex]\displaystyle \int_0^{\frac\pi2} \sqrt{\sin(x)} \, dx = \int_0^1 \frac{\sqrt x}{\sqrt{1-x^2}} \, dx[/tex]

[tex]\displaystyle \int_0^{\frac\pi2} \frac{dx}{\sqrt{\sin(x)}} = \int_0^1 \frac{dx}{\sqrt x \sqrt{1-x^2}}[/tex]

Now replace [tex]x\to\sqrt x[/tex] :

[tex]\displaystyle \int_0^1 \frac{\sqrt x}{\sqrt{1-x^2}} \, dx = \frac12 \int_0^1 x^{-\frac14} (1-x)^{-\frac12} \, dx = \frac12 B\left(\frac34, \frac12\right) [/tex]

[tex]\displaystyle \int_0^1 \frac{dx}{\sqrt x \sqrt{1-x^2}} = \frac12 \int_0^1 x^{-\frac34} (1-x)^{-\frac12} \, dx = \frac12 B\left(\frac14, \frac12\right)[/tex]

So, the original integral (which I condense here to a double integral) is

[tex]\displaystyle \int_0^{\frac\pi2} \int_0^{\frac\pi2} \sqrt{\frac{\sin(x)}{\sin(y)}} \, dx \, dy = \frac14 B\left(\frac34, \frac12\right) B\left(\frac14, \frac12\right)[/tex]

[tex]\displaystyle = \frac14 \frac{\Gamma\left(\frac14\right) \Gamma\left(\frac34\right) \Gamma\left(\frac12\right)^2}{\Gamma\left(\frac54\right) \Gamma\left(\frac34\right)}[/tex]

[tex]\displaystyle = \frac14 \frac{\Gamma\left(\frac14\right) \Gamma\left(\frac34\right) \Gamma\left(\frac12\right)^2}{\frac14 \Gamma\left(\frac14\right) \Gamma\left(\frac34\right)}[/tex]

[tex]\displaystyle = \Gamma\left(\frac12\right)^2 = \frac{\pi}{\sin\left(\frac\pi2\right)} = \boxed{\pi}[/tex]

If [tex]f^{-1}(x)[/tex] is the inverse of [tex]f(x)[/tex], then by definition

[tex]f\left(f^{-1}(x)\right) = x[/tex]

so that

[tex]2 \times 3^{f^{-1}(x)} = x[/tex]

Solve for [tex]f^{-1}(x)[/tex] :

[tex]3^{f^{-1}(x)} = \dfrac x2[/tex]

[tex]\log_3\left(3^{f^{-1}(x)}\right) = \log_3\left(\dfrac x2\right)[/tex]

[tex]f^{-1}(x) \log_3(3) = \log_3\left(\dfrac x2\right)[/tex]

[tex]\boxed{f^{-1}(x) = \log_3\left(\dfrac x2\right)}[/tex]

Answer: See attached

Step-by-step explanation:

    Given:

y = -x² + 6x - 5

    Since this is a quadratic function, we will have a parabola.

    Because our x² value is negative, this parabola will be a "frowny face" instead of a "smiley face" like the parent function.    

    The - 5 means we shift the function 5 units to the right from the parent function.

    See attached for our graph.

    What parent function am I mentioning?

The parent function of y = -x² + 6x - 5 is y = x²

Explanation:

A full circle of radius r has a circumference of 2pi*r which is the perimeter aka distance around the circle's edge.

We divide that by 4 to get the arc length of the quarter circle and we'd get to (2pi*r)/4 = 0.5pi*r

Plug in r = 1 and pi = 3.14

0.5*pi*r = 0.5*3.14*1 = 1.57

Answer:

that isn't a number

Step-by-step explanation:

Answer:

12 credits per semester

Step-by-step explanation:

she has 92 in the end but she started off with 8 so your first step is to subtract 8 from 92 which gives you 84 once you get 84 you divide that by 12 you get 7 which means she got 12 credits in each of her 7 semeters

Answer:

pack you

pa brainliwa po

Answer:

[tex]g) \left(5x-2\right)\left(3x+2\right)[/tex]

[tex]h)2\left(8a-9\right)\left(a-2\right)[/tex]

[tex]i)3\left(21n^2+42n+16\right)[/tex]

Step-by-step explanation:

Starting with g) [tex]15^2+4x-4[/tex]

Break the expression into group

[tex]=\left(15x^2-6x\right)+\left(10x-4\right)[/tex]

Now, Factor out [tex]3x[/tex] from [tex]15x^2-6x[/tex] which is now is [tex]3x(5x-2)[/tex]

Next, Facotr out [tex]2[/tex] from [tex]10x-4[/tex] which is now is [tex]2(5x-2)[/tex]

Thus,

[tex]=3x\left(5x-2\right)+2\left(5x-2\right)[/tex]

Factor common term 5x -2

[tex]\left(5x-2\right)\left(3x+2\right)[/tex]

-------------------------------------------------------------------------------------------------------------

Next we have [tex]h)16a^2-50a+36[/tex]

Factor out common term thus we have [tex]2(8a^2-25a+18)[/tex]

Factor again: [tex]8a^2-25a+18[/tex] now turn into [tex](8a-9)(a-2)[/tex]

[tex]=2\left(8a-9\right)\left(a-2\right)[/tex]

-------------------------------------------------------------------------------------------------------------

Lastly we have [tex]i)63n^2+126n+48[/tex]

Rewrite the following:

63 as 3 * 21

126 as 2 * 42

48 as 3 * 16

[tex]63n^2+126n+48[/tex]

Now cut out common term:

[tex]=3\left(21n^2+42n+16\right)[/tex]

-------------------------------------------------------------------------------------------------------------

~lenvy~

Answer:

(g)  [tex](5x-2)(3x+2)[/tex]

(h)  [tex](16a-18)(a-2)[/tex]

(i)  [tex]3(21n^2+42n+16)[/tex]

Step-by-step explanation:

To factor a quadratic in the form [tex]ax^2+bx+c[/tex]

Question (g)

[tex]15x^2+4x-4[/tex]

[tex]\implies ac=15 \cdot -4=-60[/tex]

[tex]\implies d+e=4[/tex]

Factors of 60:  1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

Therefore, the two numbers (d and e) that multiply to -60 and sum to 4 are:

10 and -6

Rewrite [tex]4x[/tex] as [tex]+10x-6x[/tex]:

[tex]\implies 15x^2+10x-6x-4[/tex]

Factories first two terms and last two terms separately:

[tex]\implies 5x(3x+2)-2(3x+2)[/tex]

Factor out common term [tex](3x+2)[/tex]:

[tex]\implies (5x-2)(3x+2)[/tex]

Question (h)

[tex]16a^2-50a+36[/tex]

[tex]\implies ac=16 \cdot 36=576[/tex]

[tex]\implies d+e=-50[/tex]

Factors of 576:  1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, 288, 576

Therefore, the two numbers that multiply to 576 and sum to -50 are:

-32 and -18

Rewrite [tex]-50a[/tex] as [tex]-32a-18a[/tex]:

[tex]\implies 16a^2-32a-18a+36[/tex]

Factories first two terms and last two terms separately:

[tex]\implies 16a(a-2)-18(a-2)[/tex]

Factor out common term [tex](a-2)[/tex]:

[tex]\implies (16a-18)(a-2)[/tex]

Question (i)

[tex]63n^2+126n+48[/tex]

Factor out common term 3:

[tex]\implies 3(21n^2+42n+16)[/tex]

This cannot be factored any further.

Answer:

3 whole but 3.4 as a decimal

Step-by-step explanation:

Answer:

8

Step-by-step explanation:

Factors of 8: 1, 2, 4, 8

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

They do share a few factors but the greatest one shared between the two is 8

4:27

Step-by-step explanation:

just subtract 45 and 18. common sense

Answer:

Ellie spent 21 minutes after she left her house until she got back home

Answer:

D

Step-by-step explanation:

8x < 24

divide both sides by the coefficient of x

8x/8 < 24/8

x < 3

Answer: $66.00

Step-by-step explanation:

56 + 10 = 66

Answer: 23

Step-by-step explanation:

Answer:

D

Step-by-step explanation:

That expression equals :

10z+6

So, now we need to find one of the expressions that ALSO gives us 10z+6

D

This is because:

2*3z=6z

2*2z=4z

----------------------

6z+4z=10z, so we got that one so far.

NEXT

2*3=6.

We got the 6 too!

it is d

Tan(angle) = opposite/ adjacent

tan(42) = 67/x

x =67/tan(42)

X = 74.411

rounded to nearest tenth : x = 74.4

Answer:

4x - 5 - 3

Step-by-step explanation:

the 3rd is the best answer i think

Answer:

here

Step-by-step explanation:

we can do it by using the formula like

mean; efx÷n

median =n+1÷2

mode =which is more

range=highest-smallest

Answer:

Step-by-stwe can do it by using the formula like

mean; efx÷n

median =n+1÷2

mode =which is more

range=highest-smallestep explanation:

Answer:

○ [tex]\displaystyle SA = 2lw + 2lh + 2wh[/tex]

Explanation:

A rectangle has four right angles, including two pairs of congruent edges, only containing two dimensions – height and length. Therefore, going three-dimensional, a rectangular prism has three dimensions – width, height, length. With this information, you should know what the answer is.

I am joyous to assist you at any time.

Answer:

7/10

Step-by-step explanation:

Fractional form of 0.7 is 7/10.

Answer:

The area is [tex]80[/tex]

Step-by-step explanation:

rectangle in the middle; 24

both triangles on the side; 56

In total; 80

Answer:

1/16th

Step-by-step explanation:

You can write this problem as a function of time.

Let x be the amount of acetaminophen that you start off with.

Let n be the amount of 2.5 hour intervals that have passed

x*0.5^n

The reason this function works is because with each 2.5 hour interval, you multiply the total product by another 1/2.

Plugging in 4(because 4 2.5 hour intervals in 10 hours), you get:

x*0.5^4,

which is x/16.

This shows that 1/16th of the acetaminophen is left in the body after 10 hours.

The graph of f(x) that satisfies the condition of f(x)'s left sided limit at 2 being -4 and right sided limit at x = 2 being 0 is given by the first graph.

In rough terminologies, the limit from one side is when we only use one side of the function with respect to the considered point of interest (in input) to predict the value of the function at that point.

For this case, we need to find the graph of the function f(x) from the considered 4  graphs for which:

[tex]\lim_{x\rightarrow 2^{-}}f(x) = -4\\\\\lim_{x\rightarrow 2^{+}}f(x) = 0\\[/tex]

For the first graph, for the point x=2, on left from the point x = 2 lies a ray. If thats the function, we can hope that the function will continue having the same line to the right too, so the prediction for value of function on x = 2 is the value that line gives for y.

That is -4.

Thus, prediction of function from left of x = 2 is -4, or

[tex]\lim_{x\rightarrow 2^{-}}f(x) = -4[/tex]

Similar to that, from the right of the point 2, there is some yellow curve. That predicts that the function's value at point x = 2 should be 0 (the yellow curve seems like touching the x-axis where y = 0 at the point x=2.

Thus, we have:

[tex]\lim_{x\rightarrow 2^{+}}f(x) = 0\\[/tex]

So graph first is correctly pertaining such function f(x).

Second graph when refered, we see that if we predict value at x =2 from left, the yellow curve in the left will make us predict that the function would be 0, so: [tex]\lim_{x\rightarrow 2^{-}}f(x) = 0[/tex]

Thus, its wrong graph.

Third graph when refered, we see that if we predict value at x =2 from left, the green line in the left will make us predict that the function would be -2 at x = 2 (see the green point in the third graph), so: [tex]\lim_{x\rightarrow 2^{-}}f(x) = -2[/tex]

Thus, its wrong graph.

Similarly, for the fourth graph, we see that if we predict value at x =2 from left, the yellow curve in the left will make us predict that the function would be 0, so: [tex]\lim_{x\rightarrow 2^{-}}f(x) = 0[/tex]

Thus, its wrong graph.

Thus, the graph of f(x) that satisfies the condition of f(x)'s left sided limit at 2 being -4 and right sided limit at x = 2 being 0 is given by the first graph.

Learn more about one-sided limits here:

https://brainly.com/question/23625942

#SPJ1

Answer:answer a

Step-by-step explanation: got right on edge 2023

Answer:

600

Step-by-step explanation:

10x9 =90

10x3=30

8x90=720

4x30=120

720-120= 600

Answer: 6.

Step-by-step explanation: To find the mean, the first step is to add all the data together.  Here's how it should look:

2 + 3 + 6 + 6 + 7 + 8 + 8 + 8 = 48.

Next, you need to divide whatever you get after adding the data by how many numbers there are for your data.  In this case, 48 was the answer when we added all the data.  In total, you have 8 numbers in your data, so you need to divide 8 from 48.  It should look like this:

48 / 8 = 6.

Therefore, the mean for this data set is 6.

Have a great day! :)

Answer:

There are a few different scenarios

Step-by-step explanation:

FROM- THE GENIUS ANSWERER

Answer:

210

Step-by-step explanation:

First, we keep $150 on the back of our minds(or our paper).

Second, we would assume that Rose's would've designed the flyer at $150, making us determine how much flyers they bought for the remaining $42.

($192 - $150 = $42)

Now that we know all of this, we can begin.

12 x 5 is 60, so 60 cents makes 1 dollar.

So for every dollar, they can make 5 flyers.

If we do 5 x 42, we'll get the answer of 210, which is the amount of flyers she bought!

Answer:

3x+2y=24

Y=3

Substitute in the equation

3x+2(3)=24

3x+6=24

3x=24-6

3x=18

X=18/3

X=6

Answer:

1 inch

Step-by-step explanation:

We know that opposite sides of a rectangle are equal to each other.

Given that,

Perimeter = 30 in

Therefore,

u + u + 14 + 14 = 30

2u + 28 = 30

2u = 30 - 28

2u = 2

Divide both sides by 2.

u = 1 in