Evaluate the following integral (Calculus 2) Please show step by step explanation!
Answers
Answer:
[tex]\displaystyle \int \dfrac{\ln x}{x^8}\:\:\text{d}x=-\dfrac{\ln x}{7x^7}- \dfrac{1}{49x^7}+\text{C}[/tex]
Step-by-step explanation:
Fundamental Theorem of Calculus
[tex]\displaystyle \int \text{f}(x)\:\text{d}x=\text{F}(x)+\text{C} \iff \text{f}(x)=\dfrac{\text{d}}{\text{d}x}(\text{F}(x))[/tex]
If differentiating takes you from one function to another, then integrating the second function will take you back to the first with a constant of integration.
Given integral:
[tex]\displaystyle \int \dfrac{\ln x}{x^8}\:\:\text{d}x[/tex]
[tex]\textsf{Apply exponent rule} \quad \dfrac{1}{a^n}=a^{-n}[/tex]
[tex]\implies \displaystyle \int x^{-8}\ln x\:\:\text{d}x[/tex]
[tex]\boxed{\begin{minipage}{4.8 cm}\underline{Integration by parts} \\\\$\displaystyle \int u \dfrac{\text{d}v}{\text{d}x}\:\text{d}x=uv-\int v\: \dfrac{\text{d}u}{\text{d}x}\:\text{d}x$ \\ \end{minipage}}[/tex]
Using Integration by parts:
[tex]\textsf{Let }u=\ln x \implies \dfrac{\text{d}u}{\text{d}x}=\dfrac{1}{x}[/tex]
[tex]\textsf{Let }\dfrac{\text{d}v}{\text{d}x}=x^{-8} \implies v=-\dfrac{1}{7}x^{-8+1}=-\dfrac{1}{7}x^{-7}=-\dfrac{1}{7x^7}[/tex]
Therefore:
[tex]\begin{aligned}\displaystyle \int u \dfrac{dv}{dx}\:dx & =uv-\int v\: \dfrac{du}{dx}\:dx\\\\\implies \displaystyle \int x^{-8}\ln x\:\:\text{d}x & = -\dfrac{1}{7x^7}\ln x- \int -\dfrac{1}{7x^7} \cdot \dfrac{1}{x}\:\:\text{d}x\\\\& = -\dfrac{\ln x}{7x^7}+ \int \dfrac{1}{7x^8}\:\:\text{d}x\\\\& = -\dfrac{\ln x}{7x^7}+ \int \dfrac{1}{7}x^{-8}\:\:\text{d}x\\\\& = -\dfrac{\ln x}{7x^7}-\dfrac{1}{7 \cdot 7}x^{-8+1}+\text{C}\\\\& = -\dfrac{\ln x}{7x^7}- \dfrac{1}{49x^7}+\text{C}\end{aligned}[/tex]
[tex]\boxed{\begin{minipage}{4 cm}\underline{Differentiating $\ln x$}\\\\If $y=\ln x$, then $\dfrac{\text{d}y}{\text{d}x}=\dfrac{1}{x}$\\\end{minipage}}[/tex]
[tex]\boxed{\begin{minipage}{4 cm}\underline{Integrating $x^n$}\\\\$\displaystyle \int x^n\:\text{d}x=\dfrac{x^{n+1}}{n+1}+\text{C}$\end{minipage}}[/tex]
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Substitute [tex]y=\ln(x) \iff e^y = x[/tex] and [tex]dy=\frac{dx}x[/tex] to get
[tex]\displaystyle \int \frac{\ln(x)}{x^8} \, dx = \int ye^{-7y} \, dy[/tex]
Integrate by parts with
[tex]u = y \implies du = dy[/tex]
[tex]dv = e^{-7y} \, dy \implies v = -\dfrac17 e^{-7y}[/tex]
Then
[tex]\displaystyle \int u\,dv = uv - \int v\,du \\\\ \implies \int y e^{-7y} \, dy = -\dfrac17 ye^{-7y} + \frac17 \int e^{-7y} \, dy + C \\\\ ~~~~~~~~~~~~ = -\frac17 ye^{-7y} - \frac1{49} e^{-7y} + C \\\\ ~~~~~~~~~~~~ = \boxed{-\frac{\ln(x)}{7x^7} - \frac1{49x^7} + C}[/tex]
Related Questions
d) The perimeter of room is 28 m. and the height is 3.5 m. Find the area of 4 walls.
it's hurry give me answer fast please
Answers
Answer: 98
Step-by-step explanation:
Area of four walls = Curved surface area
= perimeter × height
= 28 × 3.5
= 98 m square
Please help me solve this, random answers will be removed
Answers
Answer:
m∠B ≈ 70.8°
Step-by-step explanation:
The Law of Cosines relates three sides of a triangle and the angle opposite one of them.
SetupThe law of cosines tells you ...
b² = a² +c² -2ac·cos(B)
Solving for angle B, we get ...
B = arccos((a² +c² -b²)/(2ac))
where 'a' and 'c' are the sides adjacent to the angle of interest. We want angle B, so we can fill this formula as follows:
B = arccos((13² +11² -14²)/(2·13·11))
SolutionB = arccos(94/286)
B ≈ 70.812°
__
Additional comment
Another angle can be found using the Law of Sines.
A = arcsin(sin(70.812°)×13/14) ≈ 61.281°
Then angle C is ...
C = 180° -70.812° -61.281° = 47.907°
8. The cost of renting a scooter is $40 plus $4.50 per hours. Abby wants to spend less than $75
on the scooter rental. Write and solve an inequality to find the number of hours she can rent
the scooter.
Answers
Answer:
Step-by-step explanation:
4.5x + 40 < 75
4.5x < 35
x<7.7
The measure of angle is 5. The equivalent measurement in degrees is...
Answers
Answer:
300 degrees
Step-by-step explanation:
pi should be 180 degrees so 5*180/3 = 5*60=300
hich value, when placed in the box, would result in a system of equations with infinitely many solutions?
y = -2x + 4
6x + 3y =
Answers
The value placed in the box that makes the system of equation with infinitely many solution is 12.
How to solve an infinitely many solution equation?An infinite solution has both sides equal. For example, 6x + 2y - 8 = 12x +4y - 16. If we simplify the equation we will notice both sides are equal. This means the equation has an infinitely many solution.
Hence,
y = -2x + 4
Therefore,
6x + 3y = 12
divide the equation(ii) by 3
2x + y = 4
y = -2x + 4
Therefore, both equation are equal if the the box is filled with 12. This means for the value placed in the box that makes the system of equation with infinitely many solution is 12.
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How many times do you need to divide by ten to get from 0.747 to 0.0747?
Answers
Answer:
Step-by-step explanation:
Quick Answer 1
Solution
x = 0.747 / 0.0747
When you do the division, you get 10. That means that you need to divide 0.747 by just 1 ten to get 0.0747
Answer:
Once
Step-by-step explanation:
You have to divide once because if you divide a decimal by 10, you add a zero before the decimal point
Select the correct answer from each drop-down menu. B For circle O, m In the figure, 2 C CD T - 125° and m2ABC= 55°. and 2 have measures equal to 35°.
Answers
Answer:
Step-by-step explanation:
The relevant relations here are ...
the sum of arc measures in a semicircle is 180°the sum of angles in a triangle is 180°Arc measuresThe given arc CD is part of the semicircular arc CDA. The remaining arc, DA, is the supplement of CD:
arc DA = 180° -CD = 180° -125° = 55°
Central angle AOD has the same measure, 55°. That is one of the acute angles in right triangle AOB, so the other one is the complement of 55°.
∠ABO = 90° -∠AOB = 90° -55°
∠ABO = 35°
Triangle anglesIn right triangle ABC, angle ABC is given as 55°. The other acute angle, ACB, will be the complement of this.
∠ACB = 90° -∠ABC = 90° -55°
∠ACB = 35°
In the figure, angles ABO and ACB have measures of 35°.
Quick algebra 1 question for 10 points!
Only answer if you know the answer, quick shout-out to tariqareesha2 and MrBrainly, tysm for the help!
Answers
Answer:
[tex]f(x)=-2(x-3)^2+2[/tex]
Step-by-step explanation:
To find the x-intercept, substitute 0 for y and solve for x. To find the y-intercept, substitute 0 for x and solve for y.
x-intercept(s): (2,0),(4,0)
y-intercept(s): (0,−16)
Answer:
D f(x) = -2(x - 3)² + 2
Step-by-step explanation:
The y-intercept occurs at x = 0.
A
f(x) = 2x(x + 14) - 64
f(0) = -64 Not -16
B
f(x) = (x + 4)² + 2x
f(0) = 16 Not -16
C
f(x) = (x - 16)² + 4
f(0) = 260 Not -16
D
f(x) = -2(x - 3)² + 2
f(0) = -18 + 2 = -16 <-------------------- this is it
Select the correct answer. What is the value of the limit lim x 2 (x^2+3x+1/x^3+x+1) ? A. B. 1 C. D.
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So by direct evaluation we can see that the limit when x tends to 2 is equal to 1.
How to get the value of the limit?
Here we want to get the limit:
[tex]\lim_{x \to \ 2} \frac{x^2 + 3x + 1}{x^3 + x + 1}[/tex]
First, we can try to evaluate directly in x = 2 and see if it does not generate any problem, we will get:
[tex]\lim_{x \to \ 2} \frac{x^2 + 3x + 1}{x^3 + x + 1} = \frac{2^2 + 3*2 + 1}{2^3 + 2 + 1} = 1[/tex]
So by direct evaluation we can see that the limit when x tends to 2 is equal to 1.
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I need help im bad at math
Answers
Answer: 243
Step-by-step explanation:
If you divide the 18 inch by two to get the r, you will get 9. So you should multiple the 3 and the 9 square. So it should be 3x81=243
Can somebody solve this?
Answers
Answer:
MO = 12
PR = 3
Step-by-step explanation:
The perimeter of a triangle is calculated by adding all side lengths up:
The perimeter of triangle MNO is given as 48
The perimeter of triangle PQR is given as 12
The ratio is 1/4 (simplified ratio of 12/48)
we can write the following equation using this information:
[tex]\frac{x+2}{12x} = \frac{1}{4}[/tex]
cross multiply fractions
12x = 4x + 8
subtract 4x from both sides
8x = 8
divide both sides by 8
x = 1
Now to find the measure of side lengths in question we replace x with 1
for MO = 12x
12*1 = 12
for PR = x + 2
1 + 2 = 3
If a doctor charges $500 per hour for her services, how much would it cost to hire this doctor for 45 minutes?
Answers
It will cost $375 to hire the doctor for 45 minutes
The doctor charges $500 for an hour
60 minutes makes 1 hour
60 minutes= $500
Therefore the amount that will be charged for 45 minutes can be calculated as follows
60 minutes= $500
45 minutes= x
Cross multiply both sides
60x= 500×45
60x= 22500
x= 22500/60
x= 375
Hence it will cost $375 to hire the doctor for 45 minutes
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Ramiya is using the quadratic formula to solve a quadratic equation. her equation is x = startfraction negative 3 plus or minus startroot 3 squared minus 4(1)(2) endroot over 2(1) endfraction after substituting the values of a, b, and c into the formula. which is ramiya’s quadratic equation?
Answers
The quadratic equation exists [tex]$x^{2}+3 x+2=0$[/tex].
What is the formula of the quadratic equation?If a quadratic equation exists [tex]$a x^{2}+b x+c=0$[/tex], the utilizing the quadratic equation the solutions for x will be given by
[tex]$x_{1,2}=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}$$[/tex]
The solution for x exists
[tex]$x=\frac{-3 \pm \sqrt{3^{2}-4(1)(2)}}{2(1)}[/tex]
Comparing equations (1) and (2) we get, a = 1, b = 3 and c = 2.
Therefore, the quadratic equation exists [tex]$x^{2}+3 x+2=0$[/tex].
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Answer:
One guy said it was B. but I'm going with A.
Step-by-step explanation:
Ramiya is using the quadratic formula to solve a quadratic equation. her equation is x = startfraction negative 3 plus or minus startroot 3 squared minus 4(1)(2) endroot over 2(1) endfraction after substituting the values of a, b, and c into the formula. which is ramiya’s quadratic equation?
please answer this fast
Answers
The numeric values from the function are given by:
f(a) = 4 - 2a + 6a²f(a + h) = 6a² + 12ah + 6h² - 2a - 2h + 4.[f(a + h) - f(a)]/h = 12a + 6h - 2.What is the function for this problem?The function is:
f(x) = 4 - 2x + 6x².
When x = a:
f(a) = 4 - 2a + 6a².
When x = a + h:
f(a + h) = 4 - 2(a + h) + 6(a + h)² = 6a² + 12ah + 6h² - 2a - 2h + 4.
For the last question
[f(a + h) - f(a)]/h = [6a² + 12ah + 6h² - 2a - 2h + 4 - 4 + 2a - 6a²]/h = [12ah + 6h² - 2h]/h = h(12a + 6h - 2)/h
Hence:
[f(a + h) - f(a)]/h = 12a + 6h - 2.
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Grade 12 Financial Mathematic
n-up Q:
Austin buys a house for R 1 785 000 on the 1st September 2021. The bank gives him an interest rate of
6% p.a. compounded monthly. Mr Austin makes payments of R11 000 at the end of each month.
a) What is the balance on the loan on the 31st December 2036?
b) How long will it take Mr Austin to payoff the loan?
Answers
Answer:
a)0.00
b)13year,8month,2weeks and 6day
pls help ill give brainliest
a=
x=
Answers
Answer:
a = 12
x = 5.714
Step-by-step explanation:
sf = 15 / 9
20 / (15 / 9) = 12
a = 12
sf = 21 / 6
20 / (21 / 6)
x = 5.714
identify this problem plssss
Answers
Answer:
The answer is cot(x)
Step-by-step explanation:
Greetings ![tex] \tan(x) \csc {}^{2} (x) - \tan(x) [/tex]
Rewrite using trig identity
[tex] - \tan(x) + \csc {}^{2} (x) \tan(x) [/tex]
use the Pythagorean idea
[tex] \csc {}^{2} (x) = 1 + \cot {}^{2} (x) [/tex]
[tex] - \tan(x) + (1 + \cot {}^{2} (x)) \tan(x) [/tex]
simplify
[tex] \cot {}^{2} (x) \tan(x) [/tex]
use basic trigonometric identity
[tex] \tan(x ) = \frac{1}{ \cot(x) } [/tex]
simplify
[tex] \cot {}^{2} (x) \frac{1}{ \cot(x) } [/tex]
gives you
[tex] \cot(x) [/tex]
Hope it helps!
G
Which point is located at (-3.5,-4.5)?
543
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♦K
R
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point A
S
(21
te
3
21
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2 3
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157
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Answers
Answer:
Point A
Step-by-step explanation:
See attached worksheet.
What is the solution to the system of equations?
y = A system of equations. y equals StartFraction one-half EndFraction x minus 6. x equals negative 4.x – 6
x = –4
Answers
The solution to the system of equations y = 1/2x - 6, x = -4 is x = -4 and y = -8. (x, y) = (-4, -8)
EquationThere are different types of equation. Namely;
Linear equationQuadratic equationSimultaneous equationy = 1/2x - 6
x = -4
Substitute x = -4 intoy = 1/2x - 6
y = 1/2(-4) - 6
= -4/2 - 6
= -2 - 6
y = -8
Therefore,
The solution to the system of equations y = 1/2x - 6, x = -4 is x = -4 and y = -8. (x, y) = (-4, -8)
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Two cars raced at a race track. the faster car traveled 20 mph faster than the slower car. in the time that the slower car traveled 165 miles, the faster car traveled 225 miles. if the speeds of the cars remained constant, how fast did the slower car travel during the race? distance (mi) rate (mph) time (h) slower car 165 r startfraction 165 over r endfraction faster car 225 r 20 startfraction 225 over r 20 endfraction 55 mph 60 mph 75 mph 130 mph
Answers
If the faster car is travelling 20 mph faster then slower car then the speed of slower car is 55mph and faster car is 75 mph.
Given that the faster car is travelling 20 mph faster than slower car and when slower car travels 165 miles the faster car travels 225 miles.
We are required to find the speed of slower car and faster car.
let the speed of slower car be x mph.
In this way the speed of faster car be (x+20) mph.
Speed=Distance/ time
We have to first take slower car.
x=165/Time
Time=165/x-----------1
Now by taking faster car.
x+20=225/time
Time=225/(x+20)-------2
From equation 1 an equation 2.
165/x=225/(x+20)
225x=165x+3300
225x-165x=3300
60x=3300
x=55 mph
Faster car=55+20
=75 mph.
Hence if the faster car is travelling 20 mph faster then slower car then the speed of slower car is 55mph and faster car is 75 mph.
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can someone pls explain how to do this
Answers
The polynomial A is represented as x - 2, making option A the right choice.
In the question, we are asked to find the correct representation of A, when the algebraic expression (x² + x)/(x + 3) can be rewritten as the algebraic expression A + 6/(x + 3), where A is a polynomial.
Thus, we can write an equation,
A + 6/(x + 3) = (x² + x)/(x + 3),
or, A + 6/(x + 3) - 6/(x + 3) = (x² + x)/(x + 3) - 6/(x + 3) {Subtracting 6/(x + 3) from the both sides of the equation}
or, A = (x² + x)/(x + 3) - 6/(x + 3) {Simplifying},
or, A = (x² + x - 6)/(x + 3) {Subtracting the two fractions with the same denominator (x + 3)},
or, A = (x² + 3x - 2x - 6)/(x + 3) {Mid-term factorization},
or, A = {x(x + 3) - 2(x + 3)}/(x + 3) {Taking common},
or, A = {(x - 2)(x + 3)}/(x + 3) {Taking (x + 3) common},
or, A = x - 2 {Cancelling (x + 3) from the numerator and the denominator}.
Thus, the polynomial A is represented as x - 2, making option A the right choice.
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An ANOVA procedure is applied to data obtained from four distinct populations. The samples, each comprised of 15 observations, were taken from the four populations. The degrees of freedom for the numerator and denominator for the critical value of F are ________.
Answers
The degrees of freedom for the numerator and denominator for the critical value of F are _3 and 56, respectively_.
The samples, each comprised of 15 observations, were taken from the four populations.
How to find the degrees of freedom?The determine the degrees of freedom for the numerator and denominator for the critical value of F is given below:
k = 4
n = 15
Total degree of freedom;
= nk - 1
= 59
For numerator, it is
= k -1
= 4 - 1
= 3
For denominator it is
= T - (k -1 )
= 59 - 3
= 56
The degrees of freedom for the numerator and denominator for the critical value of F are _3 and 56, respectively_.
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Geometry question need help finding the answer.
Answers
Answer: 1208.96
Step-by-step explanation:
The volume of the cylinder is [tex](\pi)(4^2)(4)=64\pi[/tex]
The volume of the prism is [tex](6)(12)(14)=1008[/tex]
So, the total volume is [tex]1008+64\pi=\boxed{1208.96}[/tex]
When followers' needs for______ are strong, it makes them vulnerable to the dictates of abusive leaders
Answers
When followers' needs for security and certainty, belonging, and feeling chosen or special are strong, it makes them vulnerable to the dictates of abusive leaders.
The epistemic attribute of beliefs that one cannot rationally reject is certainty, often known as epistemic certainty or objective certainty. Epistemic certainty is commonly defined as a belief being certain if and only if the person holding it could not be wrong in holding it.
In both public and private markets, securities are fungible, tradeable financial instruments used to raise capital.
The three main categories of securities are: equity, which gives holders ownership rights; debt, which is effectively a loan returned with recurring payments; and hybrids, which include features of both debt and equity.
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Solve:
8(3 + a) = 64
A =
Answers
Answer: 8
Step-by-step explanation:
PEMDAS
if 1+sin² =3sin *cos, prove that tan = 1 or 1/2
Answers
Step-by-step explanation:
hif 1+sin² =3sin *cos, prove that tan = 1 or 1/2An art gallery wants to display 4 pieces of art in the front window. if there are 8 pieces to choose from, how many distinct displays are possible?
Answers
Answer:
70
Step-by-step explanation:
8 choose 4: 8!/(4!*4!)=70
Answer:
1680
Step-by-step explanation:
the order matters so 8P4
The lines shown below are perpendicular. If the green line has a slop of 2/5, what is the slope of the red line?
Answers
The slope of the red line that is perpendicular to the green line is: -5/2.
What are the Slope Values of Perpendicular Lines?When one line lies perpendicular to another line, the slope of one must be the negative reciprocal of the other line.
What is the Negative Reciprocal of a Number?If given a number, i.e. a/b, the negative reciprocal of a/b would the opposite value of the reciprocal of a/b.
Reciprocal of a/b is b/a. Negative reciprocal of a/b would therefore be: -b/a.
Given that the slope of the green line is: 2/5. And it is perpendicular to the red line. The slope of the red line would be the negative reciprocal of 2/5.
Negative reciprocal of 2/5 is -5/2.
Therefore, the slope of the red line is: -5/2.
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Please help!! 100 points
The graph shows a system of inequalities.
Which point is a solution to the system
(-1,6)
(0,22)
(2,9)
(8,2)
Answers
Answer: (2,9)
Step-by-step explanation:
The point lies in the region that is shaded by both inequalities.
The following duties need to be carried out: clean the board, arrange
the tables, sweep the floor and clear the waste paper basket. Find the
number of ways of assigning these duties to 4 students, given that each
student will perform only one duty?
Answers
Step-by-step explanation:
every single person will do 1/4 of each job.
divide into 2 group and each person will do half of one job and half of another.
