Answers
[tex]\bold{\huge{\underline{ Solution }}}[/tex]
- I have used different colours for indicating different process
- I used brown colour to indicate steps
- I used green colour for explaining solution
- Pink colour for known properties of integration
- Purple colour for the middle steps that are optional.
[Note :- For solving such questions it is mandatory that you should know all the functions and it's derivatives ]
Required Answer for the given question[tex]\sf{=}{\sf{\dfrac{ x^{2}}{2}}}{\sf{ + 7x + 67x .In| x - 4 | - 30.In| x - 3 | + c }}[/tex]
Answer:
[tex]\displaystyle \large{\frac{x^2}{2}+7x+67\ln |x-4|-30\ln |x-3| + C}[/tex]
Step-by-step explanation:
To find an integration of this function, first, you must know these integration methods and differences of them.
Partial Fraction Method - A method that separates the denominator into two brackets and solve the equation for two variables.Long Division Method - A method that uses long division to rewrite the fraction to make it easier to integrate.These methods above are common when it comes to integrating fractional functions, except there are differences when to use these methods.
Partial Fraction technique has to be used when the degree of numerator is lower than the degree of denominator. Basically, proper fraction.Examples (of these integrands that require partial fraction technique)
[tex]\displaystyle \large{\int \frac{3}{x^2-5x+6} \ dx}\\\displaystyle \large{\int \frac{x+2}{x^2-4} \ dx}[/tex]
Long Division technique has to be used when the degree of numerator is greater than the degree of denominator.Examples (of these integrands that require long division)
[tex]\displaystyle \large{\int \frac{x^2+5x+6}{x+2} \ dx}\\\displaystyle \large{\int \frac{x^4+3}{x} \ dx }[/tex]
Therefore, from the given integral, the integrand requires long division method. (See attachment for long division.)
After long division, we should get:
[tex]\displaystyle \large{\int x+7+\frac{37x-81}{x^2-7x+12} \ dx}[/tex]
Recall important properties of integral such as:
[tex]\displaystyle \large{\int [f(x) \pm g(x)] \ dx = \int f(x) \ dx \pm \int g(x) \ dx}[/tex]
Hence:
[tex]\displaystyle \large{\int x \ dx + \int 7 \ dx + \int \frac{37x-81}{x^2-7x+12} \ dx}[/tex]
Recall important integration formula for polynomial and constant:
[tex]\displaystyle \large{\int x^n \ dx = \frac{x^{n+1}}{n+1} + C}\\\displaystyle \large{\int x \ dx = \frac{x^2}{2} + C}\\\displaystyle \large{\int k \ dx = kx + C \ \ \tt{(k \ \ is \ \ a \ \ constant.)}[/tex]
Therefore:
[tex]\displaystyle \large{\frac{x^2}{2}+7x+\boxed{\int \frac{37x-81}{x^2-7x+12} \ dx}}[/tex]
From the boxed integral above, we cannot evaluate it by default. From what I said, if the degree of numerator is less than the degree of denominator, we’ll use partial fraction technique.
Therefore, factor the denominator:
[tex]\displaystyle \large{\frac{37x-81}{(x-4)(x-3)}}[/tex]
Set to A/x-4 and B/x-3
[tex]\displaystyle \large{\frac{37x-81}{(x-4)(x-3)} = \frac{A}{x-4} + \frac{B}{x-3}}[/tex]
Multiply both sides by (x-4)(x-3).
[tex]\displaystyle \large{\frac{37x-81}{(x-4)(x-3)} \cdot (x-4)(x-3)= \frac{A}{x-4} \cdot (x-4)(x-3) + \frac{B}{x-3} \cdot (x-4)(x-3)}\\\displaystyle \large{37x-81= A(x-3) + B(x-4)}\\\displaystyle \large{37x-81= Ax-3A+Bx-4B}\\\displaystyle \large{37x-81= (A+B)x-(3A+4B)}[/tex]
Then compare the coefficients:
[tex]\displaystyle \large{\left \{ {{A+B=37} \atop {3A+4B=81}} \right}[/tex]
Solve the simultaneous equation for A and B.
[tex]\displaystyle \large{\left \{ {{A=37-B} \atop {3A+4B=81}} \right}\\\displaystyle \large{\left \{ {{A=37-B} \atop {3(37-B)+4B=81}} \right}\\\displaystyle \large{\left \{ {{A=37-B} \atop {111-3B+4B=81}} \right}\\\displaystyle \large{\left \{ {{A=37-B} \atop {111+B=81}} \right}\\\displaystyle \large{\left \{ {{A=37-B} \atop {B=81-111}} \right}\\\displaystyle \large{\left \{ {{A=37-B} \atop {B=-30}} \right}\\\displaystyle \large{\left \{ {{A=67} \atop {B=-30}} \right}\\[/tex]
Therefore, A = 67 and B = -30. Substitute the values in:
[tex]\displaystyle \large{\int \frac{67}{x-4}-\frac{30}{x-3} \ dx}\\\displaystyle \large{\int \frac{67}{x-4} \ dx -\int \frac{30}{x-3} \ dx}[/tex]
Recall the integration formula for above:
[tex]\displaystyle \large{\int \frac{k}{x-a} \ dx = \int k \cdot \frac{1}{x-a} \ dx = k\ln |x-a| + C}[/tex]
Therefore:
[tex]\displaystyle \large{\int \frac{37x-81}{x^2-7x+12} \ dx = \int \frac{67}{x-4}-\frac{30}{x-3} \ dx = 67\ln |x-4| - 30\ln |x-3|}[/tex]
Back from this:
[tex]\displaystyle \large{\frac{x^2}{2}+7x+\boxed{\int \frac{37x-81}{x^2-7x+12} \ dx}}[/tex]
Substitute in:
[tex]\displaystyle \large{\frac{x^2}{2}+7x+67\ln |x-4|-30\ln |x-3| + C}[/tex]
Therefore the solution is:
[tex]\displaystyle \large \boxed{\frac{x^2}{2}+7x+67\ln |x-4|-30\ln |x-3| + C}[/tex]
Related Questions
Shoes sold at the Shoe Express are discounted 25% off the original price. If a pair of shoes originally cost $46, calculate the sale price of the shoes
Answers
Answer:
$34.50
Step-by-step explanation:
$46*.25 = $11.50
$46 - 11.50 = $34.50
Solution:
We know that:
Discounted price: Original price - (Original price × Discount)Original cost of a pair of shoes: $46Discount: 25%This means that:
Sale price of a pair of shoes: 46 - (46 × 25%)Step-1: Convert "25%" into a fraction.
25% = 25/100 = 1/4Step-2: Find the discounted price.
Sale price of a pair of shoes: 46 - (46 × 1/4)Sale price of a pair of shoes: 46 - (23/2)Sale price of a pair of shoes: 46 - 11.5Sale price of a pair of shoes: $34.5Susan has $65 in her savings account. Her friend coco has &119 in her savings account. Susan plans to add $23 per week. How many weeks until both friends have the same amount saved ?
Answers
Answer:
After 10 weeks, Susan will have $295 and CiCi will have $259.
Step-by-step explanation:
Susan: y= 65+23x
CiCi: y = 119+14x
65+23(6)=203
119+14(6)=203
After 6 weeks, Susan and CiCi will both have $203 in their accounts.
65+23(10)=295
119+(14)(10)=259
Therefore, After 10 weeks, Susan will have $295 and CiCi will have $259.
If Jose earned 145.60 for 7 hours, how many hours would he have to work to earn $83.20?
Answers
Answer:
4 hours
Step-by-step explanation:
Each hour: 145.60/7 = 20.80
83.20/20.80 = 4
An electric utility worker wants to anchor a guy wire from the top of a 13.2-foot utility pole to a spot that is 5.5 feet from the base of the pole on level ground. How long does the guy wire need to be?
Answers
Answer:
a^2+b^2=c^2
Step-by-step explanation:
13.2^2+5.5^2=c^2
204.49=c^2
sqrt(204.49)=14.3
RJ has two loans. Loan H has a nominal rate of 5.68%, compounded daily. Loan I has a nominal rate of 6.33%, compounded monthly. Which loan's effective rate had the greater increase, relative to its nominal rate, and how much greater is its increase than that of the other loan?
a.
Loan I’s increase was 0.03 percentage points greater than Loan H’s.
b.
Loan I’s increase was 0.68 percentage points greater than Loan H’s.
c.
Loan H’s increase was 0.16 percentage points greater than Loan I’s.
d.
Loan H’s increase was 0.49 percentage points greater than Loan I’s.
Answers
Answer:
The correct answer for this question is:
The loan's effective rate that had the greater increase, relative to its nominal rate, and how much greater is its increase than that of the other loan is that 'loan H’s increase was 0.16 percentage points greater than Loan I’s.
Step-by-step explanation:
Answer:
A: Loan I’s increase was 0.03 percentage points greater than Loan H’s.
If we divide the polynomial P by the factor x - c and we
obtain the equation P(x) = (x – c)Q(x) + R(x), then we say
that x - c is the divisor, Q(x) is the
and R(x) is
the
Answers
Hello
Step-by-step explanation:
hello I'm Filipino people
In a quadrilateral OABC, D is the midpoint of BC and E is the point on AD such that AE : ED=2: 1. Given that OA = a, OB = b and OC = c, express OD and OE in terms of a, b and c.
Answers
Answer:
Step-by-step explanation:
CB=OB-OC;
CD=CB/2=(OB-OC)/2=(b-c)/2;
OD=OC+CD=c+(b-c)/2=c/2+b/2;
AD=OD-OA=c/2+b/2-a;
AE=2/3*AD=2/3*(c/2+b/2-a)=c/3+b/3-2*a/3;
OE=OA+AE=a+c/3+b/3-2*a/3=a/3+b/3+c/3;
-------------------------------------------------------------------------------------------------
OD=c/2+b/2;
OE=a/3+b/3+c/3;
The value of OE = (a+ b+ c)/3 and OD = (b + c)/2.
In quadrilateral OABC, we are given that D is the midpoint of BC,
and E is a point on AD such that AE : ED = 2 : 1.
Also, OA = a, OB = b and OC = c
Now, By the Midpoint Theorem, we also have
OD = (1/2)OB + (1/2)OC.
OD = (1/2)(OB + OC)
OD = (1/2)(b + c)
OD = (b + c)/2.
and, CD= (OB- OC)/2 = (b-c)/2
Now, AD=OD-OA= (c+b)/2 - a
and, AE=2/3 x AD
= 2/3 x (c+ b)/2 - a
= (c + b - 2a)/3
Then, OE = OA + AE
= a + (c + b - 2a)/3
= (a+ b+ c)/3
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HELPP QUICK PLEASE please
Answers
Hi! I hope you're enjoying your day!
Please remember that an arithmetic sequence is a sequence where we use either addition or subtraction to get to the next term.
Now, let's find an arithmetic sequence here.
Is it Option A?
Remember: It should have a common difference of 2.
Which means we either add or subtract 2 to get to the next term.
Well, it seems like Option A is right.
Let's check the other options.
Is Option B right? Nope.
Here's why.
Hint: Pay attention to the signs.
We have
1, -3, 5, -7, 9...
So Option B is wrong.
Option C is correct, because we subtract 2 each time.
Option D is wrong, because we multiply by 2 each time.
Hence, the right options are
[tex]\boxed{\boxed{\bold{Options~A~and~C.}}}[/tex]
Hope everything is clear.
If you have any questions, please comment.
#LearningIsFun
[tex]\rule{200}{2}[/tex]
Question 7 of 10
What is the solution to this equation?
3(4x+3) = 2x - 5(3 - x) + 2
4
O A. x=-
15
10
O B. x=-22
O
5
O x-
O c. x - 22
15
O x
O D. x = -1
5
Answers
Answer:
Option BStep-by-step explanation:
Given Equation :
[tex] \qquad \sf \dashrightarrow \: 3(4x+3) = 2x - 5(3 - x) + 2[/tex]
Using distribute property:
[tex]\qquad \sf \dashrightarrow \: 12x + 9 = 2x - 15 + 5x + 2[/tex]
Adding the like terms we get :
[tex]\qquad \sf \dashrightarrow \: 12x + 9 = 2x + 5x - 15 + 2[/tex]
[tex]\qquad \sf \dashrightarrow \: 12x + 9 = 7x - 13[/tex]
Transposing the variables on the right side and constant terms on the left side :
[tex]\qquad \sf \dashrightarrow \: 12x - 7x = - 13 - 9[/tex]
[tex]\qquad \sf \dashrightarrow \: 5x = - 22[/tex]
Dividing both sides by 5 :
[tex]\qquad \sf \dashrightarrow \: \dfrac{5x}{5} = \dfrac{ - 22}{5} [/tex]
[tex]\qquad \bf \dashrightarrow \: x = \dfrac{ - 22}{5} [/tex]
Can you help with this question I will give you 10 points
Answers
Answer:
The elevation of University Peak could be 14,464, which in between the two mountains.
Answer:
It's shorter than Mt. Massive but it's shorter than Mt. Alverstone.
The number between 14,500 and 14,428 is 14464, that would be a great guess
Translate the triangle.
Anyways but I’ve been on this question for about 20 minutes because I can’t find the answer anywhere.
Answers
Answer:
A: (2,2) B: (3,-1) C: (-1,0)
Step-by-step explanation:
Change the coordinates of each point using the given translation. So 2 left 3 up.
Wendy works at a kite shop. Yesterday she made 2 large kites and one small kite, using 16 square yards of nylon fabric. The day before, she used 9 square yards of nylon fabric to make 1 large kite and 1 small kite. How much nylon does Wendy need to make each size kite?
Write a system of equations to describe the situation and solve using elimination. (use x for a large kit and y for a small kite)
Answers
Answer:
x=7 and y =2.
2x+y = 16
x+y = 9
Suppose that y varies directly as the square root of X, and that y = 45 when x = 225. What is y when x = 127 Round your answer to two decimal places if
necessary
Answers
Answer:
y=381
Step-by-step explanation:
given: y=45 x=225
y=kx
45=k√(225)
45=k(15)
45=15k
45/15=15k/15
3=k
given: x=127 k=3
y=kx
y=3(127)
y=381
Find the value of x, y and z.
Answers
Answer:
Step-by-step explanation:
Help will be much appreciated...
Answers
Answer:
conclusion statement four sentences about you and exothermic heat was realised or absorbed it was changed beacsue positive 9r negative.
When x=5, the value of the expression 20/-25+x-2(x-10)is
Answers
Does this represent a linear relationship or
exponential relationship?
Input Output
3 5
4 7
5 9
6 11
Answers
Answer:
It represents a linear relationship
Step-by-step explanation:
Linear relationship:
- constant change in x and y
Exponential relationship:
- constant change in x
- constant ratio of y
We see that 3, 4, 5, and 6 are a constant change in x as it's going up by +1.
It also has a constant change of y +2.
What is the equation of the line that passes through the point (2,3) and has a slope that is undefined?
Answers
Answer:
x=2
Step-by-step explanation:
An undefined slope is the slope of a vertical line.
line through (2,3) is y=mx+b
Answer: x=2
here a subtraction problem that has been solved incorrectly 21-9=11 choose the addition sentence that check this subtraction problem here the choices are 21+9=30 11+9=20 11+21=32 which one is correct
Answers
Answer:
Answer:
11+ 9 = 20
Step-by-step explanation:
=> 21-9=11
21 - 9 equals to 11 is actually a wrong answer
So in order to find the subtraction problem that will help us truly know whether the equation is wrong
=> 11+ 9 = 20
OR THE SECOND EQUATION IS
=> 21 - 11 = 10
21% of U.S. adults say they are more likely to make purchases during a sales tax holiday. You randomly select 10 adults. Find the probability that the number of adults who say they are more likely to make purchases during a sales tax holiday is (a) exactly two, (b) more than two, and (c) between two and five, inclusive.
Answers
Answer:
[tex]\left(a\right)\ 21.066\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(b\right)\ 66.872\\ \left(c\right)\ 81.567[/tex]
Step-by-step explanation:
[tex]\left(a\right)\ P\left(2\right)=C^{\begin{matrix}2\\10\\\end{matrix}}\bullet(32)^2\bullet(1-32)^{10-2}\\ =C^{\begin{matrix}2\\10\\\end{matrix}}\bullet(32)^2\bullet(68)^8=0.21066=21.066[/tex] [tex]percent[/tex]
[tex]cb)\ P(x > 2)=P(x\le2)=1-(P(x=0)+P(x=1)+P(x=2))\\ =1-C^{\begin{matrix}0\\10\\\end{matrix}}(0.32)degrees(0.68)^{10}-C^{\begin{matrix}1\\10\\\end{matrix}}(0.32)\prime(0.68)^9-C^{\begin{matrix}1\\0\\\end{matrix}}(008)(66)^9\\ =1-0.02114-0.09948-0.21066\\ =0.66872=66.872\\[/tex]
[tex](C)\ \ \ \ \ P(2\preccurlyeq[/tex] [tex]5)=P(2)+P(3)+P(4)+P(5)5)=P(2)+P(3)+P(4)+P(5)[/tex]
[tex]=C^{\begin{matrix}2\\10\\\end{matrix}}(0.32)^{)^2}\bullet(0.68)^3+C^{\begin{matrix}3\\10\\\end{matrix}}(0.32)^3\bullet(0.68)^7+C^{\begin{matrix}4\\10\\\end{matrix}}(0.32)^4\bullet(0.68)^6+C^{\begin{matrix}5\\10\\\end{matrix}}\bullet(0.32)^5\bullet(0.65)^5)[/tex]
[tex]=0.2186+0.26436+0.21771+0.12294\\ =0.81567=81.567 percent[/tex]
[tex](binomial\ distribution)[/tex]
I hope this helps you
:)
The table shows Avery’s net worth statement.
Net Worth Statement. Item; Value. Bank accounts; $3,900; Car (current value); question mark; Credit card debt; negative $2,950; Real estate; $37,425; Student loans; negative $1,700; Investments; $4,600.
Avery’s net worth is $53,755. Based on the information in the table, what is the current value of Avery’s car?
Answers
The current value of Avery’s car is $12,480.
Current valueFirst step is to calculate the Net Worth
Bank accounts $3,900
Credit card debt ($2,950)
Real estate $37,425
Student loans ($1,700)
Investments $4,600
Net worth $41,275
Second step is to calculate the current value of Avery’s car
Current value=$53,755-$41,275
Current value=$12,480
Inconclusion the current value of Avery’s car is $12,480.
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Use trigonometric rates to find the missing angle.
Answers
Because the two sides are the hypotenuse and adjacent of the angle, use cosine to figure out x! cosine is CAH (cosine= adjacent/hypotenuse) so you would do cosx-1=8/17 on the calculator which is around 62 degrees.
Answer:
Step-by-step explanation:
Each equation below is followed by several stories.
Select all of the stories that can be represented by the equation.
If none of the stories can be represented, select "None of the above".
(a)x+11=44
Answers
Typically, there are keywords in the story that tell us what operations we need to use.
For instance, in terms of age:
"Younger" or "older" connotes to subtraction from the age of the older person. It can also connote to the addition to the age of the younger person."Years ago" typically connotes to the subtraction of x years from the person's current age."More years" typically connotes to the addition of x years to the person's current age.Solving the QuestionLet's take a look at our first option.
Yousif is x years old. Rana is 44 years older than Yousif. Rana is 11 years old.Firstly, this option doesn't even make sense. How can Rana be 11 years old but also 44 years older than Yousif?
Let's look at our second option.
Rana is x years old. in 11 more years, she will be 44 years old.Immediately, we see "11 more years". This means that we will be adding 11 to Rana's current age, which is x:
[tex]x+11[/tex]
This expression above is equal to 44, since the question tells us that Rana will be 44 years old in 11 years:
[tex]x+11=44[/tex]
This is our final expression. It is the same as the given expression in the question.
AnswerRana is x years old. in 11 more years, she will be 44 years old.
What is 49 times 69 equal to?
Answers
Answer:
3381 is the answer
Answer:
3,381
Step-by-step explanation:
hope this helps
How can you calculate the area of the circle if know the circle’s circumference.
Answers
Answer:
dont cheat
Step-by-step explanation:
Z is the centroid of UVW
Z is the intersection of the triangle's ______
A. Medians
B. altitudes
C. angle bisectors
D. perpendicular bisectors
Answers
Answer: A
Step-by-step explanation:
The intersection point of the medians of a triangle is defined to be the centroid.
The solution is Option A.
The centroid of the triangle UVW is the intersection of the medians of the triangles
What is a Triangle?A triangle is a plane figure or polygon with three sides and three angles.
A Triangle has three vertices and the sum of the interior angles add up to 180°
Let the Triangle be ΔABC , such that
∠A + ∠B + ∠C = 180°
The area of the triangle = ( 1/2 ) x Length x Base
For a right angle triangle
From the Pythagoras Theorem , The hypotenuse² = base² + height²
if a² + b² = c² , it is a right triangle
if a² + b² < c² , it is an obtuse triangle
if a² + b² > c² , it is an acute triangle
Given data ,
Let the triangle be represented as UVW
Now , the median of the triangle from U is given by
The measure of WY = The measure of YV
The median of the triangle from W is given by
The measure of UX = The measure of XV
And , the point in which the three medians of the triangle intersect is known as the centroid of a triangle. It is also defined as the point of intersection of all the three medians
So , median is a line that joins the midpoint of a side and the opposite vertex of the triangle
Hence , the point Z is the intersection of the triangle's medians
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Can someone please help me out?
Answers
Answer:
Step-by-step explanation:
The function H
gives the height, above the ground, of a point on a spinning wheel. The function A gives the height of a point on a different spinning wheel and is defined by
A(t)=1.2H(3t). Here, t is time in seconds since the wheels started moving. How does the spinning wheel modeled by function A compare to the wheel for H
? Explain how you know.
Answers
The expression [tex]A(t) = 1.2\cdot H(3t)[/tex] multiplies the amplitude of [tex]H(3t)[/tex] by 1.2.
Functional analysis for the model of the height of the wheel
The model of the height is represented by the following sinusoidal model:
[tex]H(3t) = D \cdot \sin [B(3t)+E][/tex] (1)
Where:
[tex]D[/tex] - Amplitude[tex]B[/tex] - Angular frequency[tex]E[/tex] - Phase angleThen, the expression [tex]A(t) = 1.2\cdot H(3t)[/tex] multiplies the amplitude of [tex]H(3t)[/tex] by 1.2. [tex]\blacksquare[/tex]
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You move up 5 units and down 3 units. You end at (2, 0). Where did you start?
Answers
Answer:
(0,0)
Step-by-step explanation:
Do the opposite. (2-5+3,0) is (0,0)
a box of cookies weighing 4kg was divided into 6 equal shares.
Answers
Answer:
0.67
Step-by-step explanation:
that is 4 divided by 6
your answer is 0.6666666
it's a recurring decimal
or 0.67