Integrate the following:
[tex]\displaystyle \int \frac{x^2}{x^2+x+3}\, dx[/tex]
Answers
Answer:
[tex]\int {\frac{x^2}{x^2+x+3} } \, dx = - \frac{5\sqrt{11} }{11}arctan(\frac{\sqrt{11}(2x+1) }{11} ) - \frac{1}{2}ln|x^2+x+3| +x + C[/tex]
General Formulas and Concepts:
Pre-Algebra
Distributive PropertyAlgebra I
Completing the SquareRearranging VariablesAlgebra II
Long DivisionCalculus
U-Substitution[Integration Trick 1] Numerator Split[Integration Trick 2] Completing the SquareIntegration Rule 1: [tex]\int {cf(x)} \, dx = c\int {f(x)} \, dx[/tex]Integration Rule 2: [tex]\int {f(x)+g(x)} \, dx =\int {f(x)} \, dx + \int {g(x)} \, dx[/tex]Integration 1: [tex]\int {\frac{1}{u} } \, du =ln|u| + C[/tex]Integration 2: [tex]\int {\frac{du}{u^2+a^2} } = \frac{1}{a} arctan(\frac{u}{a} )+C[/tex]Integration 3: [tex]\int {x^n} \, dx = \frac{x^{n+1}}{n+1} +C[/tex]Step-by-step explanation:
Step 1: Define
[tex]\int {\frac{x^2}{x^2+x+3} } \, dx[/tex]
Step 2: Simplify Function
We do long division to simplify the function inside the function.
See Attachment for Long Division Work.
Once we do long division, our function becomes [tex]1-\frac{x+3}{x^2+x+3}[/tex]
Now we rewrite our Integral: [tex]\int ({1-\frac{x+3}{x^2+x+3} }) \, dx[/tex]
Step 3: Integrate Pt. 1
Distributive Integral [Int Rule 1]: [tex]\int {1} \, dx - \int {\frac{x+3}{x^2+x+3} } \, dx[/tex]Integrate 1st Integral [Int 3]: [tex]x - \int {\frac{x+3}{x^2+x+3} } \, dx[/tex]Step 4: Identify Variables Pt.1
Set variables for u-substitution.
u = x² + x + 3
du = (2x + 1)dx
Step 5: Integrate Pt. 2
Rewrite Integral [Int Rule 1]: [tex]x - \frac{1}{2} \int {\frac{2(x+3)}{x^2+x+3} } \, dx[/tex]Distribute 2 [Alg]: [tex]x - \frac{1}{2} \int {\frac{2x+6}{x^2+x+3} } \, dx[/tex]Rewrite Integral [Alg]: [tex]x - \frac{1}{2} \int {\frac{2x+1+5}{x^2+x+3} } \, dx[/tex]Rewrite Integral [Int Trick 1]: [tex]x - \frac{1}{2} [\int {\frac{2x+1}{x^2+x+3} } \, dx + \int {\frac{5}{x^2+x+3} } \, dx ][/tex](2nd Int) Complete the Square: [tex]x - \frac{1}{2} [\int {\frac{2x+1}{x^2+x+3} } \, dx + \int {\frac{5}{(x+\frac{1}{2})^2 + \frac{11}{4} } } \, dx ][/tex]Step 6: Identify Variables Pt. 2
Set variables for u-substitution for 2nd integral.
z = x + 1/2
dz = dx
a = √(11/4)
Step 7: Integrate Pt. 3
[Integrate] U-Substitution: [tex]x - \frac{1}{2} [\int {\frac{1}{u} } \, du + \int {\frac{5}{z^2 + (\sqrt{\frac{11}{4}})^2} } \, dz ][/tex]Rewrite Integral [Int Rule 1]: [tex]x - \frac{1}{2} [\int {\frac{1}{u} } \, du + 5\int {\frac{dz}{z^2 + (\sqrt{\frac{11}{4}})^2} } ][/tex]Integrate 1st Integral [Int 1]: [tex]x - \frac{1}{2} [ln|u| + 5\int {\frac{dz}{z^2 + (\sqrt{\frac{11}{4}})^2} } ][/tex]Integrate 2nd Integral [Int 2]: [tex]x - \frac{1}{2} [ln|u| + 5(\frac{1}{\sqrt{\frac{11}{4}}}arctan(\frac{z}{\sqrt{\frac{11}{4} } } ) ) ][/tex]Distribute 5 [Alg]: [tex]x - \frac{1}{2} [ln|u| + \frac{5}{\sqrt{\frac{11}{4}}}arctan(\frac{z}{\sqrt{\frac{11}{4} } } ) ][/tex]Distribute -1/2 [Alg]: [tex]x - \frac{1}{2}ln|u| - \frac{5}{2\sqrt{\frac{11}{4}}}arctan(\frac{z}{\sqrt{\frac{11}{4} } } )[/tex]Rationalize [Alg]: [tex]x - \frac{1}{2}ln|u| - \frac{5\sqrt{11} }{11}arctan(\frac{z}{\sqrt{\frac{11}{4} } } )[/tex]Resubstitute variables [Alg]: [tex]x - \frac{1}{2}ln|x^2+x+3| - \frac{5\sqrt{11} }{11}arctan(\frac{x+\frac{1}{2} }{\sqrt{\frac{11}{4} } } )[/tex]Simplify/Rationalize [Alg]: [tex]x - \frac{1}{2}ln|x^2+x+3| - \frac{5\sqrt{11} }{11}arctan(\frac{\sqrt{11}(2x+1) }{11} )[/tex]Rewrite [Alg]: [tex]- \frac{5\sqrt{11} }{11}arctan(\frac{\sqrt{11}(2x+1) }{11} ) - \frac{1}{2}ln|x^2+x+3| +x[/tex]Integration Constant: [tex]- \frac{5\sqrt{11} }{11}arctan(\frac{\sqrt{11}(2x+1) }{11} ) - \frac{1}{2}ln|x^2+x+3| +x + C[/tex]And we have our final answer! Hope this helped you on your Calculus Journey!
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hope this will help you........
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Related Questions
What is the coefficient of q in the sum of these two expressions?
(23q – 34) and (–16q – r)
A. 2/3
B. 3/4
C. 1/2
D. 1
Answers
answer:
1/2 bc 2/3-1/6 is equal to 4/6-1/6=3/6.
3/6=1/2
Step-by-step explanation:
love the oikawa pfp
The coefficient of {q} in the sum of the given expressions is 7.
What is expression?A mathematical expression is a combination of terms both constants and variables. For example, we can write -
2x + 3y + 5
4x + 5
Given are the expressions as -
(23q – 34) and (–16q – r)
We can write the sum of the given expressions as -
(23q – 34) + (–16q – r)
(23q - 16q) - r - 34
7q - r - 34
Here -
(7)q - r - 34
(7) is the coefficient of {q}.
Therefore, the coefficient of {q} in the sum of the given expressions is 7.
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Find the measure of <1
Answers
Consider the triangle (in which angle4 is present)
by angle sum property,
20 + 40 + angle4 = 180
60 + angle4 = 180
angle4 = 180 - 60
angle4 = 120
Also,
angle2 = 120 (vertically opposite angles)
angle1 + angle2 = 180 (linear pair)
angle1 + 120 = 180
angle1 = 180 - 120
angle1 = 60
5y=120 and 2x=108 FIND THE MISSING VALUES
Answers
To solve these, we will divide on both.
[tex]5y = 120\\\\y = 24[/tex]
[tex]2x = 108\\\\x = 54[/tex]
Name the three points shown by black dots on the graph below as ordered pairs. Put a comma and a space between each ordered pair.
Answers
Answer:
(10, 40), (15, 60), (25, 100)
Step-by-step explanation:
(x, y) = (10, 40) means x = 10, y = 40
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Answers
Answer:y=-1/4x-2
Step-by-step explanation:
Helllllpppp ASAP!!!!! Figure S is the result of a transformation on Figure R. Which transformation would accomplish this?
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Answer:
d
Step-by-step explanation:
Alejandro uses the steps below to convert StartFraction 5 over 8 EndFraction to a percent. Five-eighths right arrow 8.0 divided by 5 = 1.6. 1.6 = 160 percent. Which best explains Alejandro’s error? The division was computed incorrectly. The division was completed in the incorrect order. The decimal was moved to the right too many times. The decimal was moved to the right instead of the left.
Answers
Answer:
A
Step-by-step explanation:
Answer:
B
Step-by-step explanation:
"The output is 5 less than twice the input
Answers
Answer:
x=2x-5
Im pretty sure that's it
HELP!!!
What is the equation of this line?
A. Y = 9/15x + 21
B. Y = 2x + 9
C. Y = 2/3x + 9
D. Y = -2x - 21
Answers
Answer:
C. Y = 2/3x + 9
Step-by-step explanation:
I believe that is the answer because I graphed it. Although I could be incorrect.
can some tell me what does the tooth fairy do with the bones that come out of my face
Answers
Answer:
The tooth does very creepy stuff I have heard that she gives our teeth to the dentist for dental purposes, or even eats our teeth. I am disgusted by this already so this is your answer.
Step-by-step explanation:
Work out the value of x! Will award brainiest
Answers
Answer:
X=67.5
Step-by-step explanation:
90+X +3x =360
4x=360-90
X=270/4=67.5
15PTS PLEASE HELP ASAP!
(dont write random answers pls)
look at the pic attached:
Answers
Answer:
-1.75
Step-by-step explanation:
We know that the midpoint formula is [for the x-coordinate per say] x = 1/2x1 + 1/2x2. If we want to find a point that lies 1/4th of the way we use x = 3/4x1 + 1/4x1. How? Because the point lies 1/4th from one endpoint and 3/4th from the other.
x = 3/4x1 + 1/4x2,
x = 3/4(-3) + 1/4(2) = -9/4 + 2/4 = -7/4
= - 1.75
--- There is an easier way to do it though, but it is really vague so you should not use it. We can form a triangle with MN being the hypotenuse. What would the length of the legs be? They would be 6 and 5. Now let's form a triangle 1/4th of that. 6/4 = 1.5 and 5/4 = 1.25. If you look at the diagram, start from point N move up 1.5 units and over 1.25 forming our mini triangle. You can see the x-axis has a value of -1.75.
Caroline is working two summer jobs, Making $19 per hour lifeguarding and making $10 per hour washing cars. In a given week she could work no more than 15 total hours and was earn a minimum of $200. Also she must work a minimum of eight hours lifeguarding and a minimum of two hours washing cars, write and solve a system of inequalities graphically And determine one possible solution
Answers
Answer:
12 hours
Step-by-step explanation:
3 hours landscaping = $36
She still needs to make $224 to meet her requirement.
$224 divided by $19 tutoring = 11.79 hours
224÷19=11.79
Rounded to the nearest whole number: 12
This would mean Makayla worked a total of 15 hours and made $264. Which meets her requirements.
Which one is the answer ?
Answers
a rectangle has vertices (0,0), (2k,0) and (0,k) find the fourth vertex
Answers
Since it is at the same x-axis as (2k,0) and the same y-axis as (0,k), that is the answe
If a rectangle has vertices (0,0), (2k,0), and (0,k) the fourth vertex will be (4,8).
What is a rectangle?It is defined as two-dimensional geometry in which the angle between the adjacent sides is 90 degrees. It is a type of quadrilateral.
It is given that the rectangle has vertices (0,0), (2k,0), and (0,k).
A(0,0)
B(2k,0)
C(0,k)
D(x,y)
Since the opposite of the rectangle is the same it will be opposite to one of the coordinates shown in the graph.
To uniquely identify points on a plane, use ordered pairs and the rectangular coordinate system (x,y). Pairs that are ordered show position in relation to the origin.
The x-coordinate shows the left and right sides of the origin. Y-coordinate positions are above or below the origin.
Thus, if a rectangle has vertices (0,0), (2k,0), and (0,k) the fourth vertex will be (4,8).
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Find the UNIT RATE:
10 miles: 5 hours
A.2 hours: 1 mile
B.1 hour : 2 miles
C.2 miles: 1 hour
D.1 mile: 2 hours
Answers
Answer:
c
Step-by-step explanation:
Answer:
C
Step-by-step explanation:
3(x + 2) = -5 - 2(x - 3) *
Answers
Answer:
x = -1
Step-by-step explanation:
3(x + 2) = -5 -2(x-3)
open parenthesis
3 * x + 3 * 2 = -5 + -2 * x + -2 * -3
3x + 6 = - 5 - 2x + 6
subtract 6 from both sides
3x + 6 - 6 = -5 - 2x + 6
3x = -5 -2x
add 2x from both sides
3x + 2x = -5 - 2x + 2x
5x = -5
divide both sides of the equation by 5
5x/5 = -5/5
x = -5/5
x = -1
Click on photo if neccessary
Answers
help plsssssssss. cm.
Answers
Hope this helps
The Dubois family has $5000 set aside for an upcoming vacation in the family plans to spend 1/4 of this amount on gasoline for the drive and motel rooms along the way and 1/2 of the remainder on meals how much does the Dubois family spend on meal
Answers
Answer:
$1875
Step-by-step explanation:
Money available with Dubois family = $5000
Money to be spent on gasoline and motel room = [tex]\frac{1}{4}[/tex] of the total amount
Therefore,
Money to be spent on gasoline and motel room = [tex]\frac{1}{4} \times 5000 = \$1250[/tex]
Remaining amount of money = Total amount - Money spent on gasoline and motel room
Remaining amount of money = $5000 - $1250 = $3750
As per question statement:
Half of the remaining money is to be spent on meals.
i.e. Money to be spent on meals = [tex]\frac{1}{2}\times \$3750 = \bold{\$1875}[/tex]
Therefore, the answer is:
$1875
if f(x)=9x-2 find the value of expression f(5)
Answers
Answer:
f(x)=43
Step-by-step explanation:
you put the 5 in the places where the x is so
F(5)=9(5)-2
45-2=43
Find the slope of the line that passes through each pair of points.
C(0, –2), D(3, -2)
Answers
Answer:
0
Step-by-step explanation:
y2-y1/x2-x1
-2+2/3-0
0/3
= 0
Dianne went to the grocery store and bought 4 pounds of chicken at $3 per pound, 5 cans of green beans at $0.88 each and 2 pounds of potatoes at $1.80 per pound. If she paid with a . $50 bill, how much change will Dianne have?
Answers
Answer:
30 dollars is the amount of change she will have
Answer:
$30 is the amount of change she will have.
Step-by-step explanation:
A proportional relationship between y and x. When x = 3, y=−4 .
What is the constant of proportionality, k?
Enter your answer as a simplified or improper fraction in the box.
k =
Answers
Answer:
k = 4/3
Step-by-step explanation:
Formula for k: y/x
If y = 4 and x = 3, then we use the formula (y/x) in order to find the constant of proportionality:
k = 4/3
Heather's desk is 3 feet long. About how long is it in meters? Use 1 foot = 0.305
Answers
Answer:
0.9144 or 0.9 (rounded)
Step-by-step explanation:
0.3048*3= 0.9144 (0.9 rounded up)
or
0.305*3= 0.915 (0.9 rounded up)
Hope this helped! (ﾉ◕ヮ◕)ﾉ*:･ﾟ✧
Have a nice day! ( ﾟдﾟ)つ Bye
WHAT IS THE RATE OF CHANGE HELP!!
Answers
Answer:
B) Y=85x
Step-by-step explanation:
The actual rate of change is 75 so you need to find which one(s) have a greater one. The other 2 that are equations have a less rate of change so it is B.
Hope this helps. Pls give brainliest.
A company makes steel rods shaped like cylinders. Each rod has a radius of 2 centimeters and height of 70 centimeters. How much steel will the company need to make 243 rods? Use 3.14 for pie, and round your answer to one decimal
Answers
Answer:
213,645.6cm³
Step-by-step explanation:
Area of a cylinder is expressed as;
V = πr²h
r is the radius = 2cm
h is the height = 70cm
Get the volume of one rod;
V = πr²h
V = (3.14)(2)²(70)
V = (3.14)(4)(70)
V = 879.2cm³
one steel = 879.2cm³
243 rods = x
Cross multiply;
x = 879.2 * 243
x = 213,645.6
The company will need 213,645.6cm³ volume of steel to make 243 rods
I WILL GIVE BRAINLIEST TT
Answers
Answer:
2
Step-by-step explanation:
Yeah its 2 I can explain but will be long but if want comment and I will post but most likely its 2
I need help with this question please and thank you
Answers
d-13=5 is actually suppose to be solved liek this:
d - 13 = 5
+13 +13
The absoulte value of 13 is 13, it is negative so you have to add 13 to get to 0 so it can be cancelled out.
find the length of x and y
Answers
Answer:
x = 10.296
y = 5.831
Step-by-step explanation:
You can use the Pythagorean theorem to solve the triangles shown in various ways.
9²+5²=x²
x = 10.296
5²+3²=y²
y = 5.831