Answers
Answer:
Given Integral is:
[tex] \\ { \displaystyle{ \int{ \large{ \rm{ \frac{1}{x( {x}^{3} + 1) }dx }}}}} \\ [/tex]
Can be written as:
[tex] \\ { \large{ \longrightarrow{ \displaystyle{ \int { \rm{\frac{3 {x}^{2} }{3 {x}^{2} \times x( {x}^{3} + 1) } dx}}}}}}[/tex]
[tex] \\ { \large{ \longrightarrow \frac{1}{3} \: { \displaystyle{ \int{ \rm { \frac{ {3x}^{2} }{ {x}^{3} ( {x}^{3} + 1)} dx}}}}}}[/tex]
Now to evaluate this integral , we use the method of Substitution.
So, Substitute:
[tex] \: \: \: \: \: \: \: \: \: \: \: \: { \rightarrow \: {\large{\rm{ {x}^{3} = y}}}}[/tex]
[tex] \: \: \: \: \: \: \: \: \: \: \: \: \rightarrow \: { \large{ \rm{3 {x}^{2} dx = dy}}}[/tex]
So, on Substituting the values in above integral, we get:
[tex] \\ { \large{ \longrightarrow{ \frac{1}{3} \: \displaystyle{ \int{\rm{ \frac{1}{y(y + 1)} }}}}}}[/tex]
[tex] \\ { \large{ \longrightarrow{ \frac{1}{3} \: { \displaystyle{ \int{ \rm{ \frac{1 + y - y}{y(y + 1)}dy }}}}}}}[/tex]
[tex] \\ { \large{ \longrightarrow{ \frac{1}{3} \: { \displaystyle{ \int{ {\rm{ \frac{y + 1 - y}{y(y + 1)}dy }}}}}}}}[/tex]
[tex] \\ { \large{ \longrightarrow{ \frac{1}{3} \: { \displaystyle{ \int{ \rm{ \left[ \frac{1}{y} - \frac{1}{y + 1} \right ] }dy}}}}}}[/tex]
[tex]{ \large{ \longrightarrow{ \rm{ \frac{1}{3} \left( log |y| - log |y + 1| \right) + c}}}}[/tex]
[tex]{ \large{ \longrightarrow{ \rm{ \frac{1}{3} log | \frac{y}{y + 1} | + c }}}}[/tex]
[tex]{ \large{ \longrightarrow{ \rm{ \frac{1}{3} log | \frac{ {x}^{3} }{ {x}^{3} + 1 } | + c }}}}[/tex]
Hence,
[tex] \\{ \boxed{ \pink { \large{ \displaystyle{ \int{ \rm{ \frac{dx}{x ({x}^{3 } + 1 )} = \frac{1}{3 } log{ \huge{ |}} \frac{ {x}^{3} }{ {x}^{3} + 1} { \huge{ | }} + c}}}}}}}[/tex]
After applying algebraic substitution, partial fraction decomposition and integration rules the integral of the non-trascendental expression [tex]\frac{1}{x\cdot (x^{3}+1)}[/tex] is equal to the trascendental expression [tex]I = \frac{1}{3}\cdot \ln \left|\frac{x^{3}}{x^{3}+1} \right| + C[/tex].
How to determine the indefinite integral of a given expressionAccording to the statement we have an integrable rational expression formed by non-trascendental expressions, that is, expressions whose behavior is described solely by algebraic components. This expression can be solved by algebraic substitution after making the following modifications:
[tex]\int {\frac{dx}{x\cdot (x^{3}+1)} } = \int {\frac{3\cdot x^{2}}{3\cdot x^{3}\cdot (x^{3}+1)} } \, dx = \frac{1}{3}\int {\frac{3\cdot x^{2}}{x^{3}\cdot (x^{3}+1)} } \, dx[/tex]
And the following substitutions are used: u = x³, du = 3 · x² dx
[tex]\frac{1}{3} \int {\frac{du}{u \cdot (u+1)} }[/tex]
This is expression is equivalent to the following by partial fraction decomposition:
[tex]-\frac{1}{3}\int {\frac{du}{u+1} } \, du +\frac{1}{3}\int {\frac{du}{u} }[/tex]
And the integral is now solved:
[tex]I = -\frac{1}{3}\cdot \ln |u+1| + \frac{1}{3}\cdot \ln |u| + C[/tex]
[tex]I = \frac{1}{3}\cdot \ln \left|\frac{u}{u+1} \right| + C[/tex]
[tex]I = \frac{1}{3}\cdot \ln \left|\frac{x^{3}}{x^{3}+1} \right| + C[/tex]
Where C is the integration constant.
After applying algebraic substitution, partial fraction decomposition and integration rules the integral of the non-trascendental expression [tex]\frac{1}{x\cdot (x^{3}+1)}[/tex] is equal to the trascendental expression [tex]I = \frac{1}{3}\cdot \ln \left|\frac{x^{3}}{x^{3}+1} \right| + C[/tex].
To learn more on integrals, we kindly invite to check this: https://brainly.com/question/22008756 #SPJ5
Related Questions
Brody has a collection of 300 basketball and baseball trading cards. He has 180 baseball cards.
Answers
Answer:
The answer is 300 - 180= k
Step-by-step explanation:
Hope that helps!
Judy uses 9.7 pints of blue paint and white paint to paint her bedroom walls.3/5
of this amount is blue paint, and the rest is white paint. How many pints of white paint did she use to paint her bedroom walls?
Answers
Answer:
3.88 Pints
Step-by-step explanation:
First we need to find the amount of blue paint that was used. We can do that by multiplying 9.7 by 0.60 since that is 3/5 in decimal form
[tex]9.7*0.60=5.82[/tex]
Now we know how much blue paint was used, now we just need to find how much white paint was used. We can do that by subtracting our amount of blue paint from the total amount of paint used
[tex]9.7-5.82=3.88[/tex]
thus, 3.88 is the amount of white paint used
Please look at the picture and answer it. Thank you.
Answers
Answer: 140
Step-by-step explanation:
As you can see the angles are identical which means both sides are 140!
Day 5
Now, in this part, your knowledge will be tested again. To assess
your understanding of our lesson for this week, answer the following
tasks: Learning Task 7: Identify and describe the triangles according
to their sides
and angles.
Triangles
Sides
Angles
1.
2.
3.
4.
5.
In this part
we can measure our learnings in identifying and describing
Answers
Answer:
1. 3, 3
2 3, 3
3 3, 2
4 3, 3
5 3, 3
Which of these statements does the graph show? Check all that apply.
Answers
the first one and the last one.
which graph represents the solution to the inequality 2x-y>-3
Answers
Answer:
graph c
Step-by-step explanation:
because-3 is lower than y and 2x
Can someone please help me?
Answers
Answer:
[tex]\pi \times 4 \times 9 = 113 \\ 4 \times \pi = 12.5[/tex]
Directions: Each of the following sets are similar real-life situations. Solve these problems accurately using the appropriate combinatorics.
1. You are tasked to take pictures of Dan, Ben, Yla, Jun, and May taking three of them at a time. How many possible ways can you photograph them completely?
2. You are tasked to choose three of your classmates from Dan, Ben, Yla, Jun, and May to be your group mates. How many possible groups will you have?
please my Question}:‑)
Answers
[1] 60 possible ways
There are 5 people with 3 "spots."
We can use the following formula to solve for the possible ways that they could be photographed. This calls for a "permutation."
-> n is the total number of people, 5
-> r is the sample, 3, we are taking a picture of at a time
n! / (n−r)!
5! / (5−3)!
5! / (2)!
120 / 2
60
[2] 10 different groups
There are 5 people with 3 "spots."
We can use the following formula to solve for the possible combinations your group mates could end up being. This calls for a "combination."
-> n is the total number of people, 5
-> r is the sample, 3, we are taking 3 people to be a part of our group at a time
n! / (r!(n−r)!)
5! / (3!(5−3)!)
5! / (3!(2)!)
5! / (3!(2)!)
120 / 12
10
I spent $370 for a membership. It cost $50 to join and $20 a month. Write and Solve an equation for how many months I went?
Answers
Answer:
here
Step-by-step explanation:
use Gauth Math
source given:
$370 for a membership
cost to join: $50
370 - 50 = 320
Now, 20 a month
A month comprises of 4 weeks
A year comprises of 12 months
so,
12 X 20 = 240
320 - 240 = $120
remaining, $120
20 = 1 month
120 = 6 months
18 months???
I am not sure if this is correct, but hope it helps.
Write an equation to represent the diagram on the left.
Answers
Answer:
3n + 10 = 28
Step-by-step explanation:
3n + 10 = 28
-10 -10
3n = 18
÷3 ÷3
n = 6
I hope this helps!
Answer:
3n = 18
Step-by-step explanation:
3n = 28-10
3n = 18
If u simplify this further you will get n value as 6
(12.4×10
−4+4
.(1.5×10
4
)
Answers
Answer:
answer is 18096.
Step-by-step explanation:
12,4×10=124
124-(4+4)=124-8=116
(1,5×104)=156
116×156=18096
Which is the quotient of 1/4 ÷ 4 ? Use a number line to help.
Answers
Answer:
[tex]\frac{1}{16}[/tex]
Step-by-step explanation:
Given;
[tex]\frac{1}{4}\div \:4[/tex]
Solve;
[tex]\mathrm{Convert\:element\:to\:fraction}:\quad \:4=\frac{4}{1}[/tex]
[tex]=\frac{1}{4}\div \frac{4}{1}[/tex]
[tex]\mathrm{Apply\:the\:fraction\:rule}:\quad \frac{a}{b}\div \frac{c}{d}=\frac{a}{b}\times \frac{d}{c}[/tex]
[tex]\frac{1}{4}\times \frac{1}{4}[/tex]
[tex]\mathrm{Multiply\:fractions}:\quad \frac{a}{b}\times \frac{c}{d}=\frac{a\:\times \:c}{b\:\times \:d}[/tex]
[tex]=\frac{1\times \:1}{4\times \:4}[/tex]
[tex]\mathrm{Multiply\:the\:numbers:}\:1\times \:1=1[/tex]
[tex]=\frac{1}{4\times \:4}[/tex]
[tex]\mathrm{Multiply\:the\:numbers:}\:4\times \:4=16[/tex]
[tex]=\frac{1}{16}[/tex]
Number line given below:
~Learn with Lenvy~
Write each expression as a difference.
1. 7+3
2. a+5
3. a+b
Answers
Step-by-step explanation:
I’ll assume that difference means “subtraction”
If that’s the case then you just make everything -(-)
a) 7 - (-3)
b) a - (-5)
c) a - (-b)
Which equation represents a line that passes through (–9, –3) and has a slope of –6?
y – 9 = –6(x – 3)
y + 9 = –6(x + 3)
y – 3 = –6(x – 9)
y + 3 = –6(x + 9)
Answers
Answer:
y+3= -6(x+9)
Step-by-step explanation:
Slope is represented as M
Name the numbers in bracket as y1 and x1
You have to use the formula which is:y-y1=m(x-x1)
Then you substitute the numbers into the formula
y--3=-6(x--9)
when two negatives meet the turn to positive
thats how we can conclude that the answer is y+3=-6(x+9)
Translate each sentence into an equation.
15. Fifty-three plus four times
c is as much as 21
Answers
Answer:
53+4c=21 = 53 + 4 x c =21
Step-by-step explanation:
PLEASE HELP ITS DUE THIS SUNDAY
its okay if you don't know the answer
Answers
Answer:
109.25 the answer is 109.25 you can right this or 109 1/4
help me thank u alot <333
Answers
Answer:
N~U
P~V
Q~S
R~T
(im confused on the second picture, give me time to figure it out)
Step-by-step explanation:
1- The radius of a circle is equal to 12 cm. Find the length of the longest chord of the circle ?
2- Name the polygon whose perimeter is 35 cm and length of its side is 7 cm.
Please help me in this asp i need all the steps
Answers
Answer:
1. Longest chord is the diameter. Then answer is diameter is 12 and the radius is 6 CM.
What is the equation, in point-slope form, of the line that is parallel to the given line and passes through the given point?
Answers
Answer:
answer is D
Step-by-step explanation:
y + 1 = 3(x + 1)
12 people are watching the otters at the zoo. 7 people are watching the giraffes. How many more people are watching the otters?
Answers
Answer:
[tex]\displaystyle 5[/tex]
Explanation:
All you do is deduct the people watching giraffes from the total number of people at the zoo:
[tex]\displaystyle \boxed{5} = -7 + 12[/tex]
I am joyous to assist you at any time.
5 people’s are watching the otters …
Problem:12 people are watching the otters at the zoo. 7 people are watching the giraffes. How many more people are watching the otters ?
To find:How many more people are watching the otters ?
Let’s find: ( step-by-step explanation ):As question says ,→ At the zoo, 12 people are watching the otters & 7 people are watching the giraffes...
• So , you have to find how many people’s are watching otters more than the giraffes...
→ Then use , Subtraction to find How many more people are watching the otters..
So ,[tex]12 - 7 = 5[/tex]
• Therefore , 5 people’s are more peoples are watching the otters than the giraffes….
Hope this helps you... :)What is the slope of the line on the graph?
Answers
The slope is equal to [tex]\frac{y2-y1}{x2-x1}[/tex]
That means we need two points --> (0, 4) and (6,2)
Now let us plug it into our equation: [tex]\frac{2-4}{6-0} =\frac{-2}{6} =-\frac{1}{3}[/tex]
Thus the slope is -1/3
Hope that helps!
In the diagram, what is the value of x?
Answers
Answer:
i believe its C
Answer:
D
......
.....
....
.....
...
Identify entry B13
Please help
Answers
Answer:
-1
Step-by-step explanation:
The first number indicates the row number and the second number indicates the column number. Therefore B13،That is, the first row and the third column. -1
A matrix is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns. -1 is the value of B13.
What is Matrix?A matrix is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns.
The given matrix is [tex]B=\left[\begin{array}{ccc}2&5&-1\\0&-3&8\end{array}\right][/tex]
We need to find the value of B13.
In B13 the number one represents the row 1 and 3 represents the columns 3.
So the element in B13 position is third element of first row or first element of the third column.
-1 is the third element of first row and first element of the third column
Hence -1 is the B13 of the matrix.
To learn more on Matrices click:
https://brainly.com/question/28180105
#SPJ5
Liliana and Marcus are playing a word game. Liliana spells a word incorrectly and loses 18 points. Then Marcus spells a word correctly and has the opposite score. Which is the correct way to represent “the opposite of Liliana’s score is equal to Marcus’s score”?
Negative 18 = 18
Negative (negative 18) = negative 18
Negative (18) = negative 18
Answers
Answer:
Well if Liliana loses 18 points and Marcus get the correct one and gains 18 points
Step-by-step explanation:
So basically it's going to be negative because its saying what is the opposite of Liliana's core but in reverse order
the sum of 4.5 and a number K is 26.2
Answers
Answer:
Your answer should be K=21.7
You can find this by subtracting 26.2-4.5
What is the answer for this question?
Answers
Answer:
A
Step-by-step explanation:
Answer:
A
Step-by-step explanation:
I just took an assignment on this
Graph A represents the exponential equation y=4(6) and graph B represents y=6(4). which statements are true about graph a and graph b
Answers
Answer:
Step-by-step explanation:The following statements are true about exponential functions:
-The domain is all real numbers.
- The input to an exponential function is the exponent.
- The base represents the multiplicative rate of change.
The reason why the other two options are wrong are explained below:
The range of exponential functions is not always includes negative numbers; on the contrary, the range is the set of all positive real numbers.
The graph of an exponential function does not have a horizontal asymptote at x = 0; contrarily, the equation of the horizontal asymptote of the graph of is y = 0, which is the x-axis.
25 is 55% of what number
Answers
Answer:
45.45
Step-by-step explanation:
25 is 55% of What Number? - 25 is 55% of 45.45. 100% of 45.45 is 45.45, therefore 55 percent of 45.45 equals 25.
Answer:
x = 500/11
Step-by-step explanation:
look at the attached photo :)
In the diagram, △TPR ∼ △XPZ. Find the length of the altitude PY.
Answers
Given that △TPR and △XPZ are similar triangles, the length of the altitude PY is: 24.
What are Similar Triangles?Similar triangles have corresponding sides that have the same ratio, i.e. their sides are proportional.
To find PY, we need to know PZ.
Since △TPR ∼ △XPZ, therefore:
TR/XZ = TP/PZ
Substitute
10/20 = 13/PZ
PZ = (13 × 20)/10
PZ = 26
Find PY using Pythagorean theorem:
PY = √(PZ² - YZ²)
PY = √(26² - 10²)
PY = 24
Therefore, given that △TPR and △XPZ are similar triangles, the length of the altitude PY is: 24.
Learn more about similar triangles on:
https://brainly.com/question/11899908
Greetings to everyone! Can anyone be willing to help me in the word problems on Normal Distributions? I will forever be thankful and grateful to you!
Answers
Answer:
if some does this than this would happen
Step-by-step explanation: