find the indefinite integral 1/x^2-8x+39 dx
Answers
Complete the square in the denominator:
\(x^2 - 8x + 39 = x^2 - 8x + 16 + 23 = (x-4)^2 + 23\)
Then in the integral, substitute \(x - 4 = \sqrt{23} \tan(t)\) and \(dx = \sqrt{23} \sec^2(t) \, dt\).
\(\displaystyle \int \frac{dx}{x^2 - 8x + 39} = \int \frac{\sqrt{23} \sec^2(t) \, dt}{\left(\sqrt{23}\tan(t)\right)^2 + 23} = \frac1{\sqrt{23}} \int \frac{\sec^2(t)}{\tan^2(t)+1} \, dt\)
Recall that tan²(x) + 1 = sec²(x) for all x (such that cos(x) ≠ 0, anyway). Then the integral reduces to the trival
\(\displaystyle \frac1{\sqrt{23}} \int dt = \frac1{\sqrt{23}} t + C\)
and putting the result back in terms of x, we get
\(\displaystyle \int \frac{dx}{x^2 - 8x + 39} = \boxed{\frac1{\sqrt{23}} \tan^{-1}\left(\frac{x-4}{\sqrt{23}}\right) + C}\)
If you want to proceed via partial fractions, there's more work involved. We can use the complete-square expression to easily find the roots of the denominator:
\((x-4)^2 + 23 = 0 \implies (x-4)^2 = -23 \implies x - 4 = \pm i \sqrt{23} \implies x = 4 \pm i \sqrt{23}\)
Then we factorize
\(x^2 - 8x + 39 = \left(x - 4 - i\sqrt{23}\right) \left(x - 4 + i \sqrt{23}\right)\)
and the PFD would be
\(\dfrac1{x^2-8x+39} = \dfrac a{x - 4 - i\sqrt{23}} + \dfrac b{x - 4 + i\sqrt{23}}\)
Solve for the coefficients:
\(1 = a\left(x - 4 + i\sqrt{23}\right) + b\left(x - 4 - i\sqrt{23}\right)\)
\(\implies \begin{cases}a+b = 0 \\ \left(-4+i\sqrt{23}\right) a - \left(4+i\sqrt{23}\right) b = 1 \end{cases} \implies a = \dfrac i{2\sqrt{23}}, b=-\dfrac i{2\sqrt{23}}\)
Then the integral is
\(\displaystyle \int \frac{dx}{x^2-8x+39} = \dfrac i{2\sqrt{23}} \ln\left|x - 4 - i\sqrt{23}\right| - \dfrac i{2\sqrt{23}} \ln\left|x - 4 + i\sqrt{23}\right| + C\)
and we can condense the logarithms to
\(\displaystyle \int \frac{dx}{x^2-8x+39} = \dfrac i{2\sqrt{23}} \ln\dfrac{\left|x - 4 - i\sqrt{23}}{\left|x - 4 + i\sqrt{23}\right|} + C\)
Now we fight the urge to be discouraged by the presence of imaginary numbers in this result. The two antiderivatives are one and the same!
For any complex number z, the following identity holds:
\(\tan^{-1}(z) = -\dfrac i2 \ln \left(\dfrac{i-z}{i+z}\right)\)
With some rewriting, we have for instance
\(\dfrac i{2\sqrt{23}} \ln\dfrac{\left|x - 4 - i\sqrt{23}\right|}{\left|x - 4 + i\sqrt{23}\right|} = -\dfrac1{\sqrt{23}} \times -\dfrac i2 \ln \left|\dfrac{\frac{x-4}{\sqrt{23}} - i}{\frac{x-4}{\sqrt{23}} + i}\right| \\\\ = -\dfrac1{\sqrt{23}} \tan^{-1}\left(-\dfrac{x-4}{\sqrt{23}}\right) \\\\ = \dfrac1{\sqrt{23}} \tan^{-1}\left(\dfrac{x-4}{\sqrt{23}}\right)\)
Admittedly, I skip over a bunch of details, but the point is that both methods end with the same result, but the first method is much simpler to follow and execute, in my opinion.
Related Questions
Tyler made six loaves of pumpkin bread that had 1/4 cup of oil in each loaf. After he was done baking, he had 3/8 cup of oil remaining. How much did he have before baking
Answers
Answer:
1.875
Step-by-step explanation:
1/4 times 6= 3/2
3/2 + 3/8 =1.875
hope this helps!
the price of a wristwatch is $46.50 the sales tax rate is 5% to the nearest cent what is the total cost of the wristwatch including the sales tax
Answers
In order to determine the total cost of the wristwatch, first it is necessary to calculate the 5% of $46.50.
The 5% is given by:
(5/100)(46.50) = 2.325 ≈ 2.33
The toal cost is the sum of the sales tax rate an the price:
$46.50 + $2.33 = $48.83
I need help!!!!!!!!!!!!!!!!!!!
Answers
Answer:
Its not A!!!!!!!!!!!!!!!!!!!!!!!!
Step-by-step explanation:
Its not!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Please help me this is hard
Answers
Shaded Area=11*14 -5*7= 154- 35=119 sqm
Can someone please help with b it's urgent
Answers
Answer:
$490
Step-by-step explanation:
4.90x100= 490
I NEED HELP I WILL GIVE ALL POINTS AND MARK BRAINIEST PLEASE HELP ME
Answers
What is the approximate area of the unshaded region under the standard normal curve below? Use the portion of the standard normal table given to help answer the question.
Answers
Answer:
C: 0.18.
Step-by-step explanation:
The area of the unshaded region is 0.1815.
So, the option c is correct.
Given is a graph representing the cumulative density (Ф).
We need to find the area of the unshaded region under the standard normal curve.
From the given table Ф(1) is equal to 0.8413 and the value of Ф(2) = 0.9772.
We need to know the value of Ф(-2), so as the curve is symmetrical Ф(-2) = 1 - 0.9772
= 0.0228.
Therefore,
The shaded area under the curve will be = P(-2 ≤ z ≤ 1).
Area = 0.8413 - 0.0228
Area = 0.8185
Now,
Area of unshaded region = 1 - area of shaded region
= 1 - P(-2 ≤ z ≤ 1)
= 1 - 0.8185
= 0.1815
= 0.18
Hence, C is the correct answer.
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The complete question attached below:
If 5b = 6a + 16 and 9a = 7b - 20, then what are the values of a and b?
Answers
9514 1404 393
Answer:
(a, b) = (4, 8)
Step-by-step explanation:
Writing both equations in general form, we have ...
6a -5b +16 = 0
9a -7b +20 = 0
__
We can make a little array of the numbers in the equations:
\(\left[\begin{array}{cccc}6&-5&16&6\\9&-7&20&9\end{array}\right]\)
Note that the first column is repeated at the end.
Now, we can "cross multiply" and form the differences of products in adjacent columns:
d1 = (6)(-7) -(9)(-5) = 3 . . . . . . left 2 columns
d2 = (-5)(20) -(-7)(16) = 12 . . . . . middle 2 columns
d3 = (16)(9) -(20)(6) = 24 . . . . . right 2 columns
Then the solutions to these equations are the solutions to ...
1/d1 = a/d2 = b/d3
a = d2/d1 = 12/3 = 4
b = d3/d1 = 24/3 = 8
The values of a and b are 4 and 8, respectively.
_____
Additional Comment
There are a number of ways to solve linear equations. This is similar to what you would get by doing "elimination". It is also similar to Cramer's Rule and Vedic math techniques. The number of math operations is about the same*. Once you learn the technique, it is fairly fast and requires no decision-making. My other favorite method is using a graphing calculator.
__
* This method uses 6 multiplications, 3 subtractions, 2 divisions. In the general case of "elimination", where the coefficients are not "nice" (as here), you would perform 7 multiplications, 4 subtractions and 2 divisions. This method is less work when the coefficients don't cancel easily.
Thomas obtained a bank loan of k10 000 from BSP bank.He repays the money with 36% interest in one year.Calculate his installment payment he pays in one fortnight?
Answers
Thomas' installment payment that he pays in one fortnight is approximately k523.08.
To calculate Thomas' installment payment, we need to consider the principal amount (k10,000) and the interest rate (36%).
First, let's calculate the total amount to be repaid at the end of the year, including the interest. The interest is calculated as a percentage of the principal amount:
Interest = Principal × Interest Rate
= k10,000 × 0.36
= k3,600
The total amount to be repaid is the sum of the principal and the interest:
Total Amount = Principal + Interest
= k10,000 + k3,600
= k13,600
Now, we need to calculate the number of fortnights in a year. There are 52 weeks in a year, and since each fortnight consists of two weeks, we have:
Number of Fortnights = 52 weeks / 2
= 26 fortnights
To find the installment payment for each fortnight, we divide the total amount by the number of fortnights:
Installment Payment = Total Amount / Number of Fortnights
= k13,600 / 26
≈ k523.08
Therefore, Thomas' installment payment that he pays in one fortnight is approximately k523.08.
It's important to note that this calculation assumes equal installment payments over the course of the year. Different repayment terms or additional fees may affect the actual installment amount. It's always advisable to consult with the bank or financial institution for accurate information regarding loan repayment.
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x+y =783
10x+7y=6459
Answers
Answer:
x
=
326
,
y
=
457
Step-by-step explanation:
What is the exact length of HG in cms
Answers
The given triangle is a right angled triangle, therefore the rules of basic Trigonometry can be used here to find the solution.
Considering angle G, lets find sin 45°\(\qquad\displaystyle \tt \dashrightarrow \: \sin(45 \degree) = \frac{opposite \:\: side}{hypotenuse} \)
[ sin 45° = 1/√2 ]
\(\qquad\displaystyle \tt \dashrightarrow \: \frac{1}{ \sqrt{2} } = \frac{b}{x} \)
\(\qquad\displaystyle \tt \dashrightarrow \: x = b \sqrt{2} \: \: cm\)
So, the side HG is b√2 cm long
i need help please.......................................................
Answers
Answer:
252558855555335555422358236
Answer:
\(y=\frac{-1}{4}x+2\)
Step-by-step explanation:
The function can be written in point-slope form following this equation:
\(y=m(x-x_{0} )+y_{0}\)
where y and x are variables, and \(x_{0}\) and \(y_{0}\) are x and y values of any chosen point on the curve (line)
m (slope) can be calculated by the equation \(m=\frac{y_{2}-y_{1} }{x_{2}-x_{1}}\) where you need to find 2 chosen points on the line where \(x_{1}\) and \(y_{1}\) are the x and y values of the first set, and \(x_{2}\) and \(y_{2}\) are the x and y values of the second set.
solve for m and we get \(-\frac{1}{4}\)
Plug it into the equation with any set of x and y values on the line into the point-slope form equation, then you get the answer
If z = 38 and x = 110, then what is the value of y?
Answers
Answer:
First I need the whole question and second I am guessing around 76 or 72.
Step-by-step explanation:
A sequence is defined by the formula f(n+1)=f(n)-3. If f(4)=22, what is f(1)?
13
оооо
31
О 34
Answers
Answer:
Since f(n + 1) = f(n) - 3, we can also say that f(n - 1) = f(n) + 3.
By which we can say f(n - 3) = f(n) + 9
Since f(4) is 22, then f(1) must be 22 + 9, or 31.
How many 3/8 mile laps does shandra need to run to complete 6 miles?
Answers
Answer:
16 more 3/8 mile runs
Step-by-step explanation:
We can just divide 6 by 3/8
6/1 divided by 3/8
6/1 * 8/3
48/3
16
(a) The number of terms in an arithmetic progression is 40 and the last is -54. Given that the sum of the 15 terms added to the sum of the first 30 terms is zero. Calculate (1) The first term and common difference, (ii) the sum of the progression.
Answers
(i) The first term (a) is 24 and the common difference (d) is -2.
(ii) The sum of the progression is 2520.
i) Finding the first term and common difference:
Given that the number of terms in the arithmetic progression is 40 and the last term is -54, we can use the formula for the nth term of an arithmetic progression to find the first term (a) and the common difference (d).
The nth term formula is: An = a + (n-1)d
Using the given information, we can substitute the values:
-54 = a + (40-1)d
-54 = a + 39d
We also know that the sum of the first 15 terms added to the sum of the first 30 terms is zero:
S15 + S30 = 0
The sum of the first n terms of an arithmetic progression can be calculated using the formula:
Sn = (n/2)(2a + (n-1)d)
Substituting the values for S15 and S30:
[(15/2)(2a + (15-1)d)] + [(30/2)(2a + (30-1)d)] = 0
Simplifying the equation:
15(2a + 14d) + 30(2a + 29d) = 0
30a + 210d + 60a + 870d = 0
90a + 1080d = 0
a + 12d = 0
a = -12d
Substituting this value into the equation -54 = a + 39d:
-54 = -12d + 39d
-54 = 27d
d = -2
Now we can find the value of a by substituting d = -2 into the equation a = -12d:
a = -12(-2)
a = 24
Therefore, the first term (a) is 24 and the common difference (d) is -2.
ii) Finding the sum of the progression:
The sum of the first n terms of an arithmetic progression can be calculated using the formula:
Sn = (n/2)(2a + (n-1)d)
Substituting the values:
S40 = (40/2)(2(24) + (40-1)(-2))
S40 = 20(48 - 39(-2))
S40 = 20(48 + 78)
S40 = 20(126)
S40 = 2520
Therefore, the sum of the arithmetic progression is 2520.
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The parent function is given by f(x) = x^2
Choose the BEST description for f(x - 3)+2.
Answers
Option D
In parallelogram ABCD, diagonals AC and BD intersect at point E, AE = x2 – 24 , and CE=52
-
What is AC?
Enter your answer in the box.
units
Answers
Answer:
104
Step-by-step explanation
the area of a circle is represented by the equation A=πR², where r is the radius. The area is 64π M². what is the length of the diameter?
Answers
We know that the area of the circle can be calculated using the formula:
\(A=\pi r^2\)If we know that a certain circle has an area equal to 64π m², we can determine the radius of the circle as follows:
-First, replace the formula with the known area:
\(\begin{gathered} A=\pi r^2 \\ 64\pi=\pi r^2 \end{gathered}\)-Second, divide both sides of the equation by π:
\(\begin{gathered} \frac{64\pi}{\pi}=\frac{\pi r^2}{\pi} \\ 64=r^2 \end{gathered}\)-Third, apply the square root to both sides of the equal sign to determine the length of the radius:
\(\begin{gathered} \sqrt[]{64}=\sqrt[]{r^2} \\ 8=r \end{gathered}\)The radius of the circle is r=8m
The diameter of any circle is twice the radius, so that:
\(undefined\)Find the missing side
Answers
By using trigonometry, the missing sides are
Example 1: x = 16.7
Example 2: x = 3.2
Example 3: x = 23.5
Example 4: x = 9.3
Trigonometry: Determining the values of the missing sidesFrom the question we are to determine the value of the missing sides in the given triangles
We can determine the value of the missing sides by using SOH CAH TOA
Example 1
Angle = 42°
Opposite side = x
Hypotenuse = 25
Thus,
sin (42°) = x / 25
x = 25 × sin (42°)
x = 16.7
Example 2
Angle = 75°
Opposite side = 12
Adjacent side = x
Thus,
tan (75°) = 12 / x
x = 12 / tan (75°)
x = 3.2
Example 3
Angle = 36°
Hypotenuse side = x
Adjacent side = 19
Thus,
cos (36°) = 19 / x
x = 19 / cos (36°)
x = 23.5
Example 4
Angle = 53°
Opposite side = x
Adjacent side = 7
Thus,
tan (53°) = x / 7
x = 7 × tan (53°)
x = 9.3
Hence,
The missing sides are 16.7, 3.2, 23.5 and 9.3
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A ladder leaning against a house makes an angel of 74° with the ground. The foot of the ladder is 7 feet from the foot
of the house. How long is the ladder?
Answers
Answer:
Step-by-step explanation:
\(\frac{7}{l} =cos 74\\l=\frac{7}{cos 74} =7 sec74=7 sec(90-16)=7cos 16 \approx 6.73 ft\)
What is the intermediate step in the form
(x+a)^2=b as a result of completing the square for the following question
Answers
The intermediate step in completing the square is\($x^2 + 2ax + (a^2) = b - a^2 + (a^2)$\)
To complete the square for the equation \($(x+a)^2=b$\), we can follow these steps:
1. Expand the left side of the equation: \($(x+a)^2 = (x+a)(x+a) = x^2 + 2ax + a^2$\).
2. Rewrite the equation by isolating the squared term and the linear term: \($x^2 + 2ax = b - a^2$\).
3. To complete the square, take half of the coefficient of the linear term, square it, and add it to both sides of the equation:
\($x^2 + 2ax + (a^2) = b - a^2 + (a^2)$\).
4. Simplify the right side of the equation: \($x^2 + 2ax + (a^2) = b$\).
This step can be represented as: \(\[x^2 + 2ax + (a^2) = b - a^2 + (a^2)\]\)
This intermediate step helps us rewrite the equation in a form that allows us to factor it into a perfect square.
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Help please:)
Graph the equation by plotting points.
X=4
Answers
Answer:
(4,0)
Step-by-step explanation:
You basically are plotting a point on the positive number 4 on the x line. Since they're only asking for an X and not a Y, you'd leave it as (4,0). Hope this helps!
What is the value of b?
Answers
Since b + 24 and 2b are equal to each other the equation will be this :
B + 24 = 2b
You will subtract b on both sides
24 = b
This is the answer
A wage earner receives a gross pay of $983.15 for 52.5 hours of work. What is his hourly rate of pay if a regular workweek is 44 hours and overtime is paid at time-and-a-half the regular rate of pay?
Answers
$56.76
Step by Step explanation:
We must first identify the regular pay and overtime pay before we can calculate the hourly rate of pay. The wage earner worked 8.5 hours over the standard 44-hour workweek, or 52.5 hours total. We must first establish the regular rate of pay because the overtime rate of pay is 1.5 times that amount.
Let's name the standard hourly wage "x." The standard wage for 44 hours is 44 hours * x, which comes to $44x. The overtime pay would be $12.75 for 8.5 hours * x * 1.5. $44 x + $12.75 x = $56.75 x = $983.15 would be the entire salary.
So, $983.15 / $56.75x = 17.45. And, x = $983.15 / 17.45 = $56.76.
The measure of ∠1 is 39°. What is the measure of ∠2?
Answers
Answer:
141
Step-by-step explanation:
if the sum of the two angles equals 180 subtract 39 from 180 to get the remainder of 141 which is angle 2
Quinton brought 40 cupcakesfo school onhis birthday. He gave a cupcake to eachof the 18 students in Ms. Delmont's class. He also gave a cupcake to eachof the 16 students in Mrs. Donnelly's class. He also gave a cupcake to Ms. Delmont, Mrs. Donnelly, the school nurse, and the schoolprincipal. How many cupcakes did he haveleft over?
Answers
Answer:
2
Step-by-step explanation:
18+16+4=36
40-38=2
Answer:
2 cupcakes left
Step-by-step explanation:
40 - (18 + 16 + 1 + 1 + 1 + 1) = 2
Find the median for the given sample data.
The distances (in miles) driven in the past week by each of a company's sales representatives are listed below.
78 126 238 284 310 356
Find the median distance driven.
A. 261 mi
B. 238 mi.
C. 198.50 mi.
D. 284 mi.
Answers
Answer:
d = 261 miles
Step-by-step explanation:
Given that,
The distances (in miles) driven in the past week by each of a company's sales representatives are listed below.
78 126 238 284 310 356
We need to find the median of the distances.
There are 6 data. If n is even,
\(median =\dfrac{\dfrac{n}{2}^{th}+(\dfrac{n}{2}+1)^{th}}{2}\\\\median =\dfrac{\dfrac{6}{2}^{th}+(\dfrac{6}{2}+1)^{th}}{2}\\\\=\dfrac{3^{rd}+4^{th}}{2}\\\\=\dfrac{238+284}{2}\\\\=261\ mi\)
So, the median distance is 261 miles.
2 Mabaso has R140, Thabo has R70 and Ally has R35. What is the ratio of the amount of money Mabaso has, to the amount of money Thabo has and to the amount of money Ally has? Write the ratios in simplest form. The price of a steel table is R750. On Black Friday the table could be bought for R600. Calculate the percentage discount? Show ALL your calculations. Convert 125 g to kilograms. (1 kg = 1 000 grams) A green grocer packs 12 apples in a plastic bag. Calculate the number of bags he w need if he has 285 apples. The scale of a map is 1 500 000. Determine the actual distance in km if measurement on the map is 23,7 cm. Hint: 1 km = 100 000 cm
Answers
The actual distance represented by 23.7 cm on the map is 355.5 km.
To find the ratio of the amount of money Mabaso has to the amount of money Thabo has and the amount of money Ally has, we can divide each amount by the smallest amount (which is R35) to simplify the ratio.
Mabaso has R140, Thabo has R70, and Ally has R35.
The ratio of Mabaso's money to Thabo's money is:
R140 ÷ R35 = 4
The ratio of Mabaso's money to Ally's money is:
R140 ÷ R35 = 4
Therefore, the ratio of the amount of money Mabaso has to the amount of money Thabo has and to the amount of money Ally has is 4:1:1.
To calculate the percentage discount of a steel table, we need to find the difference between the original price and the discounted price, and then divide it by the original price. Finally, we multiply the result by 100 to get the percentage.
Original price: R750
Discounted price: R600
Discount: R750 - R600 = R150
Percentage discount: (R150 ÷ R750) × 100 = 20%
So, the table has a 20% discount on Black Friday.
To convert 125 grams to kilograms, we divide the amount in grams by 1,000 (since there are 1,000 grams in a kilogram).
125 g ÷ 1,000 = 0.125 kg
Therefore, 125 grams is equal to 0.125 kilograms.
If a green grocer packs 12 apples in a plastic bag and has 285 apples, we divide the total number of apples by the number of apples per bag to determine the number of bags needed.
Number of bags needed: 285 apples ÷ 12 apples/bag = 23.75 bags
Since we can't have a fraction of a bag, we round up to the nearest whole number. Therefore, the green grocer would need 24 bags.
If the scale of a map is 1,500,000 and the measurement on the map is 23.7 cm, we can use the scale to determine the actual distance.
1 cm on the map represents 1,500,000 cm in reality.
23.7 cm on the map represents x cm in reality.
x = 23.7 cm × 1,500,000 cm = 35,550,000 cm
To convert cm to km, we divide by 100,000 (since there are 100,000 cm in a kilometer).
35,550,000 cm ÷ 100,000 = 355.5 km
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Help please I will mark you as brainlyist!!!
Answers
Answer: 8
Step-by-step explanation:
Answer:
Step-by-step explanation:
12 - 6x > 24
First, add 6x to both sides:
12 - 6x + 6x > 24 + 6x
Subtract 24 from both sides:
12 - 24 > 24 + 6x - 24
-12 > 6x
Divide both sides by 6:
-12 ÷ 6 > 6x ÷ 6
-2 > x
S = (∞, -2)
Hope this helps!