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!
![Integrate The Following: [tex]\displaystyle \int \frac{x^2}{x^2+x+3}\, Dx[/tex]](https://brainimage.s3.us-east-005.backblazeb2.com/answers/attachments/IlhTHv3cfGOetUQuYJVGdsQUORcf0ghv.png)
hope this will help you........
![Integrate The Following: [tex]\displaystyle \int \frac{x^2}{x^2+x+3}\, Dx[/tex]](https://brainimage.s3.us-east-005.backblazeb2.com/answers/attachments/jMcFLsl3Su42CPIw5XPVgChWo1DsKNfN.jpeg)
Related Questions
Which pair of angles are complementary but not adjacent?
O 1 and 25
O 1and 3
O 4 and 1
O 4 and 5
Answers
????????????????????last one loll
Answers
Answer: Hope this helped you!
Step-by-step explanation:
If you start of will 10 and you eat 6 you ate 6/10 which is equal 60%
What is this i dont know pls help me
Answers
please help! What is true about the graph on the interval from point c to point d?
Answers
Planes A and B are shown. If a new line, p, is drawn parallel to line l, which
statement is true?
O Line p must be drawn in plane B.
O Line p must be perpendicular to line m.
O Line p must be drawn so that it can lie in the same plane as line l.
O Line p must be drawn in the same plane as line n
Answers
Answer: the second one
Step-by-step explanation:
I’m smart I have al a’s
The true statement is - Line p must be drawn so that it can lie in the same plane as line l.
What is plane?"It is a two-dimensional flat surface with no edges or thickness."
What are parallel lines?"Two lines in the same plane that are at equal distance from each other and never meet."
For given question,
Planes A and B are shown.
If a new line, p, is drawn parallel to line l, must be drawn so that it can lie in the same plane as line l.
Line l lie in plane A.
So, the line p must be drawn in the plane A.
Therefore, the true statement is - Line p must be drawn so that it can lie in the same plane as line l.
Learn more about parallel lines here:
https://brainly.com/question/16701300
#SPJ3
write the equation of the lines fully simplified
Answers
Answer:
y=-4x-5
Step-by-step explanation:
PLEASE HELP PLEASE EMERGENCY
Answers
Answer:First one is first step. 2 is Last step. 3 is fourth step. 4 is second step. 5 is Third step
Step-by-step explanation:
I went down the column
A real estate agent sells a house for $92,000. She receives a 6% commission on the sale of the house. How much did she earn on the sale? Round your answer to the nearest cent. Pls help i need thins in less than 5 minuets pls
Answers
$5,520.00 there ya go
rewrite y=-8x+1 in standard form
Answers
Answer:
8x+y=1
Step-by-step explanation:
Hope this helps!!
The length and width of one rectangle are 8 and 5.
The width of the other rectangle is 5, but its length is unknown so we label it `x`.
Answers
Answer:
fksfk
Step-by-step explanation:
Answer: x is 8
Step-by-step explanation:
4 T =
fl OZ
(T = Tablespoons)
Answers
Answer:
2 fl oz
Step-by-step explanation:
Answer:
4 tablespoons is = to 2 fl oz
Please help this is due right now!
Answers
Answer:
i believe its B
Step-by-step explanation:
Okay. B because your paying more than store A so B will pay the phone off faster than store A . if i am wrong let me know.
HELP PLEASE!!!! I ONLY HAVE 3 MORE MATH RIDDLES LEFT!!!!
Answers
Answer:
97
Explanation:
Buddy- 6
Hermey-3
____________________________________________________________
It takes Buddy 94 years to be 100 so you would add 94 to both boys ages!
6+94=100 years old!
3+94= 97 years old!
Answer:
He is going to be 50 years old
Step-by-step explanation:
if he was 3 when his brother was 6 then it would mean that when his brother is 100 he should be 50 because that would be half. You can divide it if you want some extra proof :)
Noah collected five water samples from local streams. Each sample was the same size, and he collected 4.35 liters of water in all. What was the volume of each water sample?
Answers
Answer:
4.35/5= 0.87 liters
you divide the total by how many to find out what the unit rate is for each sample
Step-by-step explanation:
What is the slope of a line perpendicular to the line whose equation is 12x-3y=81. Fully reduce your answer.
Answers
Answer:
-1/4
Step-by-step explanation:
-3y = 81-12x
y= 4x - 27
4m = -1 x 1/4
= -1/4
Claudia spends $2.15 day for lunch.Her cafeteria lunch account has a balance of 48.00.Which equation can be used to determine the maximum number of days,d,that she can eat without adding money to her lunch account?
Answers
Given :
Claudia spends $2.15 day for lunch.
Her cafeteria lunch account has a balance of 48.00 .
To Find :
The maximum number of days, d that she can eat without adding money to her lunch account.
Solution :
Number of days she can eat without adding money in her account is :
[tex]n = \dfrac{Remaining \ Balance }{2.15}\\\\n = \dfrac{48}{2.15}\\\\n = 22.325[/tex]
But, number of days cannot be in decimal.
Therefore, maximum number of days she can eat without adding money to her lunch is 22 days.
In 2020, the United States population is 3.31 x 108. What is this number in standard form?
А
33,100,000
B
331,000,000,000
С
331,000,000
D
3,310,000
anybody know the answer??
Answers
Answer:331,000,000
Step-by-step explanation:
someone pls explain how to do this step by step
Answers
Answer:
7
Step-by-step explanation:
12=5+4p+3p ---> subtract the 5 to the other side
12-5=4p+3p ---> combine like terms
7=p
Hope this helps! :)
I need an answer for both of these
Answers
Answer:
1. x < -2
2. x < -9
Step-by-step explanation:
1. x + 15 < 13
Minus 15 on both sides.
x + 15 - 15 < 13 - 15
+15 - 15 makes a 0.
x < -2
2. x + 11 < 2
Minus 11 on both sides.
x + 11 - 11 < 2 - 11
+11 - 11 makes a 0.
x < -9
Answer:
1. x < -2
2. x < -9
3. x < 15
Step-by-step explanation:
Treat the < and > symbols as an equals sign to make it less confusing to solve. Then solve for x like you normally would.
1. x+ 15 = 13
subtract 15 from both sides
x=-2, or x < -2
2. x + 11 = 2
Subtract 11 from both sides.
x=-9, or x <-9
3. 19 > x + 4
19 = x + 4
Subtract 4.
15 > x, or x < 15
Body temperatures that are 0.9°F above or below 98.6°F are considered normal. What equation can be used to determine the least and greatest normal temperatures? What are Ithe temperatures?
pls help fast
Answers
Answer: i got a for your answer choice
Step-by-step explanation:
what is 5x+60y=35
x+y=1.5
Answers
Solving the system of equations shows that to make these equations true x=1 and y=0.5
can someone help me with these two please , i’ll cashapp
Answers
The answer is B. 59°
4.
Part A: The measure of a is 70°.
Explanation: The angle sum in any triangle is 180°. 70 + 40 is 110. To find the final angle, subtract 110 from 180. 180 - 110 = 70
Part B: The length of x is 5m.
Explanation. The triangle is an isosceles triangle. We know this because the two angles on the base are equal. Since this is an isosceles triangle, the two longer sides are the same length. Therefore: x = 5 m.
If anyone could answer this it would be great!!
Answers
Answer:
27/8
Step-by-step explanation:
first you would convert -.5 to -1/2 and add everything up by getting the same denominator
Answer:
3.375
Step-by-step explanation:First convert the 2 1/4 and the 1 5/8
then you just add it with the 0.5 or just use a calculator
2 1/4 = 9/4
1 5/8= 13/8
okay now do this ---> 0.5 + 9/4 +13/8 =3.375
Five friends went to Madrid. The return flight was $25 more than the outgoing flight. Airport taxes were $21 for each flight for each person. The overall cost was $595. What was the price of the return flight for each person?
Answers
Answer:
$98
Step-by-step explanation:
21 × 5 ( since there are 5 friends )
= 105
595 - 105 = 490 (total cost for all 5 of the return flight)
490 ÷ 5 = 98
$ 98 for each person.
Can someone help me on this! plz help me and tell me how u got the answer.
and plz dont troll.....
Answers
Answer: The last choice, 2x + 3y = 10
Step-by-step explanation: This equation will have the same slope. The line will be parallel to the original side , just higher up on the grid.
please help someone
Answers
Answer:
m1 =131 degrees
m2=49 degrees
Step-by-step explanation:
Alternate interior angles are always congruent, so they will be equal to each other, so m1=131 degrees.
Straight angles always add up to 180 degrees, so 180-131=49, so m2=49 degrees
Abby bought one package of cookies that weighed 24 ounces and another package that weighed 2 pounds 2 ounces. What was the combined weight of the two packages of cookies she bought?
Answers
Answer: If you place 10 packages per box, each box will weigh 10(3 lb 4 oz). This is 30 lb 40 oz. For 4 boxes you would have 4(30 lb 40 oz). This is 120 lb 160 oz. To put you answer in the correct form you need to change the 160 oz to pounds and ounces. 16 oz is equal to 1 pound, so divide 160 by 16. The 160 oz is equivalent to 10 lbs. Therefore the total weight of 4 boxes is 120 + 10 which equals 130 lbs.
HELPP ASAPPPP
Evaluate the expression 10m - 11
if m=4
Answers
Answer:
is the answer 45444.2453 jd
Step-by-step explanation:
Answer:
10m -11 ( plug in the given value for m)
10 (4) -11 (do multiplication first because of PEMDAS)
40 -11 (subtract)
29 (the answer)
can someone help me
Answers
============================================================
Explanation:
Let's find the slope of the table. This will tell us the rate of change.
m = slope
m = (y2-y1)/(x2-x1)
m = (25250 - 26375)/(2 - 1) .... I used the first two rows
m = -1125/1
m = -1125
The slope of -1125 means the plane's altitude is decreasing at a rate of 1125 feet per minute. This is the descent speed.
Since we're told the initial altitude was 27500 ft, this means b = 27500
This means y = mx+b turns into y = -1125x+27500
x is the number of minutes and y is the altitude at time x
As a way to check this, plug in various x values of the table. You should get corresponding y values that match the table. I'll let you do this part.
------------------
Now plug in y = 16250 and solve for x
y = -1125x+27500
16250 = -1125x+27500
-1125x+27500 = 16250
-1125x = 16250-27500
-1125x = -11250
x = -11250/(-1125)
x = 10
It takes 10 minutes after the descent period has started, for the pilot to get to 16,250 ft.
Since the pilot was contacted at the x = 4 minute mark, this means 10-4 = 6 minutes is the duration needed for the pilot to go from 23,000 ft to 16,250 ft.
A private college allows local residents to use athletic facilities such as the gym and the pool for a fee. The college charges a one time registration fee of $25 and also charges a fee of $18 per semester of use. complete the table to show the total cost of using the athletic facilities after each semester listed.
Answers
Step-by-step explanation:
1) 43
2) 61
3) 79
4)97
5)115
Answer:
popoverlover20 is correct I got it right
Step-by-step explanation:
