(1 point) The intensity of light at a depth of x meters below the surface of a lake satisfies the differential equation dIdx=(−1.21)I. (a) At what depth, in meters, is the intensity of light half that of L, where L equals the intensity of light at the surface (where x=0)?

The function whose graph is given is y = sin (x - 2).

Option B is the correct answer.

A function has an input and an output.

A function can be one-to-one or onto one.

It simply indicated the relationships between the input and the output.

Example:

f(x) = 2x + 1

f(1) = 2 + 1 = 3

f(2) = 2 x 2 + 1 = 4 + 1 = 5

The outputs of the functions are 3 and 5

The inputs of the function are 1 and 2.

We have,

The graph of y = sin(x - 2) is a sinusoidal function that is shifted 2 units to the right from the standard sine function y = sin(x).

The sine function oscillates between -1 and 1 as x increases, and the value of x at which the function reaches its minimum or maximum value is a multiple of π.

When we subtract 2 from x in the equation y = sin(x - 2), the entire graph is shifted to the right by 2 units, which means that the minimum and maximum points occur at x-values that are 2 units greater than they would be for the standard sine function.

The graph of y = sin(x - a) is a sinusoidal function that is shifted a units to the right from the standard sine function.

So, in this case, the graph of y = sin(x-2) looks like the standard sine function shifted 2 units to the right.

The amplitude and period of the function remain the same as the standard sine function, but the phase shift changes.

Thus,

The function whose graph is given is y = sin (x - 2).

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Answer:

a)

[tex]P(X = 0) = h(0,100,15,5) = \frac{C_{5,0}*C_{95,15}}{C_{100,15}} = 0.4357[/tex]

[tex]P(X = 1) = h(1,100,15,5) = \frac{C_{5,1}*C_{95,14}}{C_{100,15}} = 0.4034[/tex]

[tex]P(X = 2) = h(2,100,15,5) = \frac{C_{5,2}*C_{95,13}}{C_{100,15}} = 0.1377[/tex]

[tex]P(X = 3) = h(3,100,15,5) = \frac{C_{5,3}*C_{95,12}}{C_{100,15}} = 0.0216[/tex]

[tex]P(X = 4) = h(4,100,15,5) = \frac{C_{5,4}*C_{95,11}}{C_{100,15}} = 0.0015[/tex]

[tex]P(X = 5) = h(5,100,15,5) = \frac{C_{5,5}*C_{95,10}}{C_{100,15}} = 0.00004[/tex]

b) 0.154% probability that there are at least 4 defective welders in the sample

Step-by-step explanation:

The welders are chosen without replacement, so the hypergeometric distribution is used.

The probability of x sucesses is given by the following formula:

[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}*C_{N-k,n-x}}{C_{N,n}}[/tex]

In which

x is the number of sucesses.

N is the size of the population.

n is the size of the sample.

k is the total number of desired outcomes.

Combinations formula:

[tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

In this question:

100 welders, so [tex]N = 100[/tex]

Sample of 15, so [tex]n = 15[/tex]

In total, 5 defective, so [tex]k = 5[/tex]

(a) Determine the PMF of the number of defective welders in your sample?

There are 5 defective, so this is P(X = 0) to P(X = 5). Then

[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}*C_{N-k,n-x}}{C_{N,n}}[/tex]

[tex]P(X = 0) = h(0,100,15,5) = \frac{C_{5,0}*C_{95,15}}{C_{100,15}} = 0.4357[/tex]

[tex]P(X = 1) = h(1,100,15,5) = \frac{C_{5,1}*C_{95,14}}{C_{100,15}} = 0.4034[/tex]

[tex]P(X = 2) = h(2,100,15,5) = \frac{C_{5,2}*C_{95,13}}{C_{100,15}} = 0.1377[/tex]

[tex]P(X = 3) = h(3,100,15,5) = \frac{C_{5,3}*C_{95,12}}{C_{100,15}} = 0.0216[/tex]

[tex]P(X = 4) = h(4,100,15,5) = \frac{C_{5,4}*C_{95,11}}{C_{100,15}} = 0.0015[/tex]

[tex]P(X = 5) = h(5,100,15,5) = \frac{C_{5,5}*C_{95,10}}{C_{100,15}} = 0.00004[/tex]

(b) Determine the probability that there are at least 4 defective welders in the sample?

[tex]P(X \geq 4) = P(X = 4) + P(X = 5) = 0.0015 + 0.00004 = 0.00154[/tex]

0.154% probability that there are at least 4 defective welders in the sample

Answer:

78doalrse

Step-by-step explanation:

solos w

Answer:

$11.40

Step-by-step explanation:

This question asks us to find how much money the family should leave as their tip.

To find the tip, multiply the cost of the total bill by the tip rate.

cost of bill * tip rate

The total bill was $57, and they are leaving a 20% tip

$57 * 20%

Convert 20% to a decimal by dividing by 100 or moving the decimal place 2 spaces to the left.

20/100=0.20

20.0--> 2.0 --> 0.20

20%=0.20

$57 * 0.20

Multiply 57 and 0.20

11.4= $11.40

The family should leave a tip of $11.40

Answer:

first answer is x= -5 second answer is y=7.

Step-by-step explanation:

7-5=2. 2+3=5.

7-1=6.

Answer:

obtuse

isosceles

Step-by-step explanation:

It has one angle bigger than 90 so it is obtuse

It has two angles that measure the same so it has two sides that measure the same so it is isosceles

The given triangle is an Obtuse triangle.

A triangle is a three sided polygon, which has three vertices and three angles which has the sum 180 degrees.

We have to given that;

All the angles are,

100°

40°

40°

Here, It has one angle bigger than 90 so it is obtuse.

And, It has two angles that measure the same so it has two sides that measure the same so it is isosceles.

Thus, The given triangle is an Obtuse triangle.

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Answer:

a-3b+4

Step-by-step explanation:

1/2 (2a−6b+8) =

=1/2*2a-1/2*6b+1/2*8

=a-3b+4

Answer:

a - 3b + 4

Explanation:

According to the instructions, we must apply the distributive property to create an equivalent expression.

* reminder

distributive property formula: a (b + c) = ab + ac

Let's start by applying the distributive property to the expression.

[tex]\displaystyle\frac{1}{2} (2a - 6b + 8)\\\\\displaystyle\frac{1}{2} (2a) + \displaystyle\frac{1}{2} (-6b) + \displaystyle\frac{1}{2} (8)[/tex]

Simplify by multiplying.

[tex]\displaystyle\frac{1}{2}(2a)+\displaystyle\frac{1}{2}(-6b)+\displaystyle\frac{1}{2}(8)\\\\a+\displaystyle\frac{1}{2}(-6b)+\displaystyle\frac{1}{2}(8)\\\\a-3b+\displaystyle\frac{1}{2}(8)\\\\a-3b + 4[/tex]

Therefore, an equivalent expression to the given expression is a - 3b + 4.

The value of X is 3.25

Look at the attached picture

Hope it will help you

Good luck on your assignment

Answer:

[tex]x=15\sqrt{5}[/tex]

Step-by-step explanation:

[tex]15 \times \sqrt{5} = x[/tex]

[tex]33.54102 \approx x[/tex]

Answer:

Step-by-step explanation:

r(t) = (7 cost + 7t sin t)i + (7 sin t - 7t cos t)j

[tex]\frac{d \bar r t}{dt} =(7\frac{d}{dt}\cos t + 7\frac{d}{dt} (t \sin t)i+(7\frac{d}{dt} \sin t-7\frac{d}{dt} t \cos t)j[/tex]

[tex]=(7(-\sin t)+7(1* \sin t+t \cos t))i+(7 \cost -7(1*\cos t - t \sin t))j\\\\=7((-\sin t+\sin t+t \cos t)i+(\cos t-\cos t+t \sin t)j)\\\\=7((t\cos t)i+(t\sin t)j)[/tex]

[tex]\bar r'(t)=\frac{d \bar r t}{dt} =(7t\cos t)i+(7t\sin t)j---(1)\\\\11\bar r(t)=\sqrt{(7t\cos t)^2+(7t\sin t)^2}\\\\=\sqrt{49t^2(\cos^2t+\sin^2 t)} \\\\=7t[/tex]

[tex]\bar T (t)=\frac{\bar r'(t)}{11\bar r(t)11} =\frac{(7t\cos t)i+(7t\sin t)j}{7t} \\\\\barT(t)=(\cos t)i+(\sin t)j[/tex]

[tex]\bar T'(t)=\frac{d}{dt} (\cos t)i+\frac{d}{dt} (\sin t) j\\\\\bar T'(t)=(-\sin t)i+(\cos t)j---(2)\\\\11\bar T'(t)=\sqrt{(-\sin t)^2+(\cos t)^2} \\\\=\sqrt{\sin^2t+\cos^2t} \\\\=1[/tex]

[tex]\bar N(t)=\bar T'(t)=\frac{(-\sin t)i+(\cos t)j}{(1)} \\\\ \large \boxed {\bar N(t)=(-\sin t)i+(\cos t)j}[/tex]

[tex]K(t)=\frac{|\b\r T'(t)|}{\bar r (t)|} \\\\=\frac{|-\sin t i+\cos t j|}{|7t\cos t +7t \sin t j|}[/tex]

Using eq (1) and (2)

[tex]K(t)=\frac{\sqrt{(-\sin t)^2+(\cos t)^2} }{\sqrt{(7t\cos t)^2+(7t\sin t)^2} }\\\\=\frac{\sqrt{\sin^2 t+\cos^2t} }{\sqrt{49t^2(\cos^2 t+\sin^2t)} }\\\\=\frac{\sqrt{1} }{\sqrt{49t^2\times 1} } \\\\ \large \boxed {K(t)=\frac{1}{7t} }[/tex]

The solution to the given function is determined as 13.

The solution of the function is calculated as follows;

[tex]\sqrt{x - 4} \ +\ 5 = 2[/tex]

collect similar terms together

 [tex]\sqrt{x - 4} \ = 2-5\\\\\sqrt{x - 4} \ = -3[/tex]

square both sides of the equation

[tex](\sqrt{x - 4} )^2 = (-3)^2\\\\x - 4 = 9\\\\x = 9 + 4\\\\x = 13[/tex]

Thus, the solution to the given function is determined as 13.

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Answer:

No solution

Step-by-step explanation:

Edge.

Answer:

99.36% probability that the sample proportion will differ from the population proportion by less than 6%

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For a sample proportion p in a sample of size n, we have that the sampling distribution of the sample proportions has [tex]\mu = p, s = \sqrt{\frac{p(1-p)}{n}}[/tex].

In this question:

[tex]n = 306, p = 0.18, \mu = 0.18, s = \sqrt{\frac{0.18*0.82}{306}} = 0.0220[/tex].

What is the probability that the sample proportion will differ from the population proportion by less than 6%

This is the pvalue of Z when X = 0.18 + 0.06 = 0.24 subtracted by the pvalue of Z when X = 0.18 - 0.06 = 0.12. So

X = 0.24

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

By the Central Limit Theorem

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{0.24 - 0.18}{0.022}[/tex]

[tex]Z = 2.73[/tex]

[tex]Z = 2.73[/tex] has a pvalue of 0.9968

X = 0.12

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{0.12 - 0.18}{0.022}[/tex]

[tex]Z = -2.73[/tex]

[tex]Z = -2.73[/tex] has a pvalue of 0.0032

0.9968 - 0.0032 = 0.9936

99.36% probability that the sample proportion will differ from the population proportion by less than 6%

Answer:

Step-by-step explanation: idk

Answer:

Step-by-step explanation:

you just do, everyone in here needs help so i dont think theres people in here just messing arround, you can trust.

Answer:

because we are amazing as always.

Step-by-step explanation:

Please dont do this or i will report u.

bye

1. So start off by asking yourself, how will I get this answer? Absolute values!

2. So what the question is asking is for you to build and equation that shows us how to solve for the highest and lowest grades that are 19 points above and below the average of 72.

3. So your unknown is x, and for us to be able to plug in our highest and lowest scores it will always have to = 19 (that’s the rule we got from the question)

So now this is where your absolute value comes in.

4. Set it up

| x - 72 | = 19

5. Solve it

72-19=53 Minimum

72+19=91 Maximum

6. | 53 - 72 | = 19

| 91 - 72 | = 19

Answer:

(a) dy/dx = 4/(2y+1)^2.

(b) y = 4/9 x - 14/9

(c) d2y/dx2 = -64/243

Step-by-step explanation:

You have the following equation

[tex](2y+1)^3-24x=-3[/tex]   (1)

(a) You first derivative implicitly the equation (1) respect to x:

[tex]\frac{d}{dx}[(2y+1)^3-24x]=\frac{d}{dx}[-3]\\\\3(2y+1)^2(2\frac{dy}{dx})-24=0[/tex]

next, you solve the last result for dy/dx:

[tex]6(2y+1)^2\frac{dy}{dx}=24\\\\\frac{dy}{dx}=\frac{4}{(2y+1)^2}[/tex](2)

(b) The equation for the tangent line is given by:

[tex]y-y_o=m(x-x_o)[/tex]    (3)

with yo = -2 and xo = -1

To find the slope m you use the result of the equation (2), because dy/dx evaluated in (-1,-2) is the slope at such point:

m = [tex]\frac{dy}{dx}=\frac{4}{(2(-2)+1)^2}=\frac{4}{9}[/tex]

Hence, by replacing in the equation (3) you obtain:

[tex]y-(-2)=\frac{4}{9}(x-(-1))\\\\y+2=\frac{4}{9}x+\frac{4}{9}\\\\y=\frac{4}{9}x-\frac{14}{9}[/tex]

hence, the equation for the tangent line is y = 4/9 x - 14/9

(c) To find d2y/dx2 you derivative the result obtain in the equation (2):

[tex]\frac{d^2y}{dx^2}=\frac{d}{dx}[4(2y+1)^{-2}]\\\\\frac{d^2y}{dx^2}=-8(2y+1)^{-3}(2\frac{dy}{dx})\\\\\frac{d^2y}{dx^2}=-16(2y+1)^{-3}\frac{dy}{dx}[/tex]     (4)

the second derivative for the point (-1,-2) is obtained by replacing y=-2 and dy/dx=m=4/9 in the equation (4):

[tex]\frac{d^2y}{dx^2}=-16(2(-2)+1)^{-3}(\frac{4}{9})=-\frac{64}{243}[/tex]

hence, d2y/dx2 evaluated in (-1,-2) is -64/243

Answer:

(A) The value of  [tex]dy/dx=\frac{4}{(2y+1)^2}[/tex].

(B) The equation of the tangent is : [tex]y=(4/9)x-(14/9)[/tex]

(C) The value of  [tex]\frac{d^2y}{dx^2}=-64/243[/tex]

(D) The point (16,0) is not on curve so it can not be determined by the given equation.

Step-by-step explanation:

Given information:

The equation [tex](2y+1)^3-24x=-3[/tex]

(A) For the first derivative of the given equation:

[tex]\frac{d}{dx}[(2y+1)^3-24x ]= \frac{d}{dx}(-3)[/tex]

[tex]3(2y+1)^2(dy/dx)-24=0[/tex]

[tex](dy/dx)=\frac{4}{(2y+1)^2}[/tex]

Hence , from the above equation it is shown that the value of

[tex]dy/dx=\frac{4}{(2y+1)^2}[/tex]

(B) The equation of the tangent to the curve is given by:

[tex]y-y_o=m(x-x_0)\\[/tex]

On putting the given values in the above equation

We get:

[tex]m=\frac{4}{(2(-2))+1)^2}[/tex]

[tex]m=4/9[/tex]

Hence, the equation of the tangent can be written as :

[tex]y-(-2)=(4/9)(x-(-1))\\y+2=\frac{4}{9}x+\frac{4}{9}[/tex]

So, the equation of the tangent is :

[tex]y=(4/9)x-(14/9)[/tex]

(C) Now ,

To find [tex]d^2y/dx^2[/tex] for the equation

We have to find double derivative of the equation;

[tex]\frac{d^2y}{dx^2} =\frac{d}{dx}[4(2y+1)^{-2}]\\\frac{d^2y}{dx^2} = -16(2y+1)^{-3}\frac{dy}{dx}[/tex]

On putting the values from the given information in the above equation;

[tex]\frac{d^2y}{dx^2} =-16(2(-2)+1)^{-3}(4/9)[/tex]

[tex]\frac{d^2y}{dx^2}=-64/243[/tex]

(D) For the equation [tex](2y+1)^3-24x=-3[/tex]

First check for the given points (16,0) if it satisfies the given equation or not.

Now on checking for the same the point is not satisfying the given equation hence, we can not find the value of [tex](y-1)'(0)[/tex].

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Answer:

3889

Step-by-step explanation:

you add all the numbers to get the answer

Answer:3889

Step-by-step explanation:So what u want to do is add them

1034

1289

1566

3889

Answer:

Persia should measure the length of each flower that she used in creating this flower arrangement and add up all the values to get the total height (i.e., the combined height) of the eight flowers used. Make sure to keep the units consistent all throughout the calculation to avoid any errors. For example, if centimetres are used to measure height of one flower, use centimetres all throughout and not switch to using inches at any point.

Hope that answers the question, have a great day!

Answer:

61 1/2

Step-by-step explanation:

math is ez

4 and 5!

4 squared is 16, which is less than 24, and 5 squared is 25, which is more than 24!

[tex]\sqrt{24}[/tex] lies between two consecutive numbers  4  and 5

Given :

Given square root of 24

Lets write all the perfect square numbers

[tex]\sqrt{4}=2\\\sqrt{9}=3\\\sqrt{16} =4\\\sqrt{25} =5\\\sqrt{36} =6\\\sqrt{49}=7[/tex]

From the above perfect square root numbers, we can see that square root (24) lies between [tex]\sqrt{16} \; and\; \sqrt{25}[/tex]

So we can say that [tex]\sqrt{24}[/tex] lies between 4  and 5

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Answer:

You cannot do this because 2800 cm squared is area and liters is volume. however, if you mean 2800 cm cubed, then the answer is 2.8 liters.

Step-by-step explanation:

Answer:

Before A. 32.7247330577, A. 27.8204757722, B. 32.7247330577, C. 15.4868329805, and D. 32.7247330577

Step-by-step explanation:

Used a calculator. Not that Hard.

Answer:

Second quadrant = 120°.

Third quadrant = 210°

Step-by-step explanation:

We are given that:

[tex]sin^2(x) = 3cos^2(x)[/tex]

The following property is known:

[tex]sin^2(x) +cos^2(x)=1\\[/tex]

Combining both expressions:

[tex]sin^2(x) =1-cos^2(x)\\sin^2(x) = 3cos^2(x)\\\\1-cos^2(x) = 3cos^2(x)\\cos^2(x)=\frac{1}{4}\\cos(x)=\pm \frac{1}{2}[/tex]

If x lies in the second quadrant, then cos(x) = -1/2:

[tex]x=cos^{-1}(1/2)\\x=120^o[/tex]

The value of x that satisfies the equation if x lies in the second quadrant is 120°.

If x lies in the third quadrant, then cos(x) = -1/2:

[tex]x=120+90=210^o[/tex]

The value of x that satisfies the equation if x lies in the third quadrant is  210°

Answer:

The value of x that satisfies the equation if x lies in the second quadrant is 120

The value of x that satisfies the equation if x lies in the third quadrant is

240

Step-by-step explanation:

This is correct for Plato/Edmentum users :) Hope I could help !

Answer:

2700000

Step-by-step explanation:

Because it is not at 750000 it gets rounded down

2. The integration region,

[tex]\left\{(r,\theta)\mid\dfrac\pi6\le\theta\le\dfrac\pi2\land2\le r\le3\right\}[/tex]

corresponds to what you might call an "annular sector" (i.e. the analog of circular sector for the annulus or ring). In other words, it's the region between the two circles of radii [tex]r=2[/tex] and [tex]r=3[/tex], taken between the rays [tex]\theta=\frac\pi6[/tex] and [tex]\theta=\frac\pi2[/tex]. (The previous question of yours that I just posted an answer to has a similar region with slightly different parameters.)

You can separate the variables to compute the integral:

[tex]\displaystyle\int_{\pi/6}^{\pi/2}\int_2^3r^2\sin^2\theta\,\mathrm dr\,\mathrm d\theta=\left(\int_{\pi/6}^{\pi/2}\sin^2\theta\,\mathrm d\theta\right)\left(\int_2^3r^2\,\mathrm dr\right)[/tex]

which should be doable for you. You would find it has a value of 19/72*(3√3 + 4π).

3. Without knowing the definition of the region D, the best we can do is convert what we can to polar coordinates. Namely,

[tex]r^2=x^2+y^2[/tex]

so that

[tex]\displaystyle\iint_De^{x^2+y^2}\,\mathrm dA=\iint_Dre^{r^2}\,\mathrm dr\,\mathrm d\theta[/tex]

Answer:

36,000

Step-by-step explanation:

So all we have to do is divide 360 by 0.01. that equals 36,000

Step-by-step explanation:

federal= $4180

(9500*0.11*4)

private=$2660

(9500*0.07*4)

Answer:

We study if the composition of both functions equals the identity ("x"), that is if

[tex]f(g(x))=x[/tex]

Step-by-step explanation:

The composition of the two functions should render 'x" if one is the inverse of the other. That is, we need to find what  [tex]f(g(x))[/tex] renders. Notice as well that the same verification could be done with examining  [tex]g(f(x))[/tex].

Let's work with [tex]f(g(x))[/tex] :

[tex]f(g(x))=f(\frac{1}{5} x+5)=5\,(\frac{1}{5} x+5)-25= x+25-25=x[/tex]

So we see that the composition of both functions indeed render "x", and that way we have verified that one is the the inverse of the other.

Answer:

B. One-fifth (5 x minus 25) + 5

Step-by-step explanation:

Just got it right on the test.

I don't see a square root sign anywhere, so I'll assume the integral is

[tex]\displaystyle\int\sqrt{13+12x-x^2}\,\mathrm dx[/tex]

First complete the square:

[tex]13+12x-x^2=49-(6-x)^2=7^2-(6-x)^2[/tex]

Now in the integral, substitute

[tex]6-x=7\sin t\implies\mathrm dx=-7\cos t\,\mathrm dt[/tex]

so that

[tex]t=\sin^{-1}\left(\dfrac{6-x}7\right)[/tex]

Under this change of variables, we have

[tex]7^2-(6-x)^2=7^2-7^2\sin^2t=7^2(1-\sin^2t)=7^2\cos^2t[/tex]

so that

[tex]\displaystyle\int\sqrt{13+12x-x^2}\,\mathrm dx=-7\int\sqrt{7^2\cos^2t}\,\cos t\,\mathrm dt=-49\int|\cos t|\cos t\,\mathrm dt[/tex]

Under the right conditions, namely that cos(t) > 0, we can further reduce the integrand to

[tex]|\cos t|\cos t=\cos^2t=\dfrac{1+\cos(2t)}2[/tex]

[tex]\displaystyle-49\int|\cos t|\cos t\,\mathrm dt=-\frac{49}2\int(1+\cos(2t))\,\mathrm dt=-\frac{49}2\left(t+\frac12\sin(2t)\right)+C[/tex]

Expand the sine term as

[tex]\dfrac12\sin(2t)}=\sin t\cos t[/tex]

Then

[tex]t=\sin^{-1}\left(\dfrac{6-x}7\right)\implies \sin t=\dfrac{6-x}7[/tex]

[tex]t=\sin^{-1}\left(\dfrac{6-x}7\right)\implies \cos t=\sqrt{7^2-(6-x)^2}=\sqrt{13+12x-x^2}[/tex]

So the integral is

[tex]\displaystyle-\frac{49}2\left(\sin^{-1}\left(\dfrac{6-x}7\right)+\dfrac{6-x}7\sqrt{13+12x-x^2}\right)+C[/tex]