Answers
Given:
[tex]\int_4^{\infty}\frac{1}{x^2+x}\text{ dx}[/tex]To find:
the integral
[tex]\begin{gathered} First,\text{ we will re-write the expression} \\ \frac{1}{x^2+x}\text{ = }\frac{1}{x^2(1\text{ + }\frac{1}{x})} \\ \\ let\text{ u = 1 + 1/x} \\ u\text{ = 1 + x}^{-1} \\ \frac{du}{dx\text{ }}\text{ = 0 + \lparen-1}x^{-1-1})\text{ = -1x}^{-2} \end{gathered}[/tex][tex]\begin{gathered} \frac{du}{dx}\text{ = -x}^{-2} \\ \\ du\text{ = -x}^{-2}dx \\ du\text{ = }\frac{dx}{-x^2} \\ \\ \int_4^{\infty}\frac{1}{x^2+x\text{ }}dx\text{ = }\int_4^{\infty}\frac{1}{x^2(1\text{ +}\frac{1}{x})}dx \\ \\ Substitute\text{ for u and du in the expression:} \\ \int_4^{\infty}\frac{1}{x^2(u)}dx\text{ = }\int_4^{\infty}\frac{dx}{-x^2(u)}=\int_4^{\infty}-\frac{du}{u} \\ \end{gathered}[/tex][tex]\begin{gathered} -\int_4^{\infty}\frac{du}{u}=-\int_4^{\infty}ln\text{ u \lparen differentiation rule\rparen} \\ \\ \int_4^{\infty}ln(1+\frac{1}{x})=\int_4^{\infty}ln(\frac{x+1}{x})=\int_4^{\infty}ln(x+1)\text{ - ln\lparen x\rparen} \\ \\ -\int_4^{\infty}ln(1+\frac{1}{x})=-\int_4^{\infty}ln(x+1)\text{ - ln\lparen x\rparen = }\int_4^{\infty}ln(x)\text{ - ln\lparen x+1\rparen} \\ \\ -\int_4^{\infty}ln(1+\frac{1}{x})=\text{ \lbrack\lparen}\lim_{x\to\infty}(ln(x)\text{ - ln\lparen x+1\rparen\rbrack- \lbrack lnx - ln\lparen x+1\rparen\rbrack}_{x=4} \\ \\ -\int_4^{\infty}ln(1+\frac{1}{x})=\text{ \lbrack}\frac{x}{x+1}\text{\rbrack}_{\infty}\text{ - ln\lbrack}\frac{x}{x+1}]_4 \\ \\ -\int_4^{\infty}ln(1+\frac{1}{x})=0\text{ - ln\lbrack}\frac{4}{4+1}] \\ \\ -\int_4^{\infty}ln(1+\frac{1}{x})=\text{ -ln\lbrack}\frac{4}{5}] \end{gathered}[/tex][tex]-\int_4^{\infty}ln(1+\frac{1}{x})\text{ = ln\lparen}\frac{5}{4})[/tex]Related Questions
Which expression is equivalent to 20 — 3(x + 2)?A 3X+ 14B —3x + 14 C -9x + 21 D 17x— 34
Answers
We have to simplify the expression:
[tex]20-3(x+2)[/tex]and see which expression is equivalent.
We can do it like this:
[tex]\begin{gathered} 20-3(x+2) \\ 20-3\cdot x-3\cdot2 \\ 20-3x-6 \\ 20-6-3x \\ 14-3x \\ -3x+14 \end{gathered}[/tex]This expression is equivalent to -3x+14.
Answer: -3x+14 [Option B]
Answer:
first option
Step-by-step explanation:
[tex]\frac{\frac{-2}{x}+\frac{5}{y} }{\frac{3}{y}-\frac{2}{x} }[/tex] ← combine fractions on numerator and denominator
= [tex]\frac{\frac{-2y+5x}{xy} }{\frac{3x-2y}{xy} }[/tex]
leave numerator, change division to multiplication and turn denominator 'upside down'
= [tex]\frac{-2y+5x}{xy}[/tex] × [tex]\frac{xy}{3x-2y}[/tex] ← cancel xy on numerator/ denominator
= [tex]\frac{-2y+5x}{1}[/tex] × [tex]\frac{1}{3x-2y}[/tex]
= [tex]\frac{-2y+5x}{3x-2y}[/tex]
Convert the following: 253 mm to m. ANS. _______ m
Answers
253mm is read 253 Millimeter.
'Milli" is a sub-multiple that has a value of:
[tex]10^{-3}[/tex]Thus, 253mm is
[tex]253\times10^{-3}m[/tex]Expressing this in standard form, it becomes:
[tex]2.53\times10^{-1}m[/tex]To Decimal places, it is;
[tex]0.253[/tex]Between 10 P.M and 7:20 A.M., the water level in a swimming pool decreased by 7/12 in. Assuming that the water level decreased at a constant rate, how much did the water level drop each hour? PLEASE HELP I DONT GET THIS AT ALL!
Answers
Answer:
0.0625 or [tex]\frac{1}{16}[/tex]
Step-by-step explanation:
Interpreting the ProblemIf the water level decreases at a constant rate, then that just means that the relationship between the water level and time is linear or is a straight line if graphed.
Calculating Constant Rate:let's just say that: [tex]C = \text{ contant rate the water level dropped at each hour}[/tex]
this means if we added C by how many hours passed, we should get the amount the water level dropped: [tex]C+C+C+C\text{...how many hours passed} =\frac{7}{12}[/tex]
let's also just say that: [tex]H = \text{ amount of hours that passed}[/tex]
from here we can rewrite the equation using multiplication: [tex]CH = \frac{7}{12}[/tex]
we can now divide both sides to isolate C: [tex]C = \frac{\frac{7}{12}}{H}[/tex], so now all we have to do is find how many hours passed.
From 10 P.M to 12 A.M, 2 hours pass. From 12 A.M to 7 A.M, 7 hours pass and from 7 A.M to 7:20 A.M, 20 minutes pass
So we have: [tex]2 \text{ hours} + 7 \text{ hours} + 20 \text{ minutes} = 9\text{ hours} + 20\text{ minutes}[/tex]
We want to represent this as one value and also in hours, so we'll need to convert the minutes to hours. To see how many 20 minutes is to one hour, we simply divide this 20 minutes by how many minutes are in an hour, which is 60 minutes: [tex]\frac{20}{60} = \frac{1}{3}[/tex]
It's actually super useful to keep this in fraction form, and even convert the 9 hours to fraction form: [tex]9 + \frac{1}{3} = \frac{27}{3} + \frac{1}{3} = \frac{28}{3}[/tex]
Now from here, we know that: [tex]H = \frac{28}{3}[/tex]
so let's plug this into the expression we made! [tex]C = \frac{\frac{7}{12}}{\frac{28}{3}}\implies \frac{7}{12} * \frac{3}{28}[/tex]
before multiplying, we can rewrite 12 as (4 * 3) so we can cancel out the 3 in the numerator and denominator making the simplification process a bit easier. We can also rewrite 28 as (4 * 7) to cancel out the 7 in the numerator and denominator: [tex]C = \frac{7 * 3}{(4 * 3) * (7 * 4)} = \frac{1}{16} = 0.0625[/tex]
This means if we multiplied the 1/16 or 0.0625 by the 9.33333 hours that passed we would get the total amount that decreased: 7/12
2. At the gas station, three small drinks and two large drinks contain 108 ounces ofcola. A small drink contains a third as much cola as a large drink. How much coladoes each size drink contain?
Answers
Let x = small drinks
Let y = large drinks
3 small drinks and 2 large drinks contain 108 ounces of cola, this is:
3x + 2y = 108
A small drink contains a third as much cola as a large drink, this is:
x = 1/3y
Then, we solve the system of equations:
[tex]\begin{gathered} 3x+2y=108 \\ x=\frac{1}{3}y \end{gathered}[/tex]First, substitute x in equation 1:
[tex]3(\frac{1}{3}y)+2y=108[/tex]And solve for y:
[tex]\begin{gathered} y+2y=108 \\ 3y=108 \\ \frac{3y}{3}=\frac{108}{3} \\ y=36 \end{gathered}[/tex]Next, substitute y = 36 in x:
[tex]x=\frac{1}{3}y=\frac{1}{3}(36)=12[/tex]Answer:
Small drinks: 12 ounces of cola
Large drinks: 36 ounces of cola
A section of a quilt is shaped like a parallelogram.What is the minimum amount of fabric that is needed to cover this section completely? A 13 Square InchesB 17 Square InchesC 21 Square InchesD 26 Square Inches
Answers
The area of a parallelogram is computed as follows:
A = b*h
where b is the base and h is the height.
From the picture, the base is: 2 + 4.5 = 6.5 inches, and the height is 4 inches. Then its area is:
A = 6.5*4 = 26 square inches
Given this super-sized board (16x16), what integer lengths are possible for slanted segments? Use the line tool to sketch them (using a different color for each one). Label each length. Then describe how you found them.
Answers
A way to find integer line segments is using Pythagorean triples, that is, positive integers that are consistent with the Pythagorean theorem, for example, (3,4,5) because we have
[tex]3^{2^{}}+4^2=5^{2^{}}[/tex]therefore, they can be put in a triangle like this
Therefore, the slanted segment would have a length of 5. That can be done with other Pythagorian triples like (5,12,13) or (8,15,17).
Evaluate the expression, writing the result as a simplified complex number.My answer 3iI know is wrong but I don’t know why.
Answers
The first thing we can do is solve the i cubed:
[tex]undefined[/tex]Write a variation equation for the following situation. Use k as the constant of variation.R varies inversely as the square of h.The variation equation is ______
Answers
Given
R varies inversely as the square of h
Find
Equation for the given statement
Explanation
[tex]\begin{gathered} R\propto\frac{1}{h^2} \\ \\ R=\frac{k}{h^2} \end{gathered}[/tex]Final Answer
The equation for given statement is
[tex]R=\frac{k}{h^2}[/tex]A project on Kickstarter for an iPad stylus raised 1,130% of their goal, raising a total of $322,507 from 7,457 supporters. What was their original goal?
Answers
Let:
x = Original goal
y = Final goal = $322507
a = Percentage raised = 1.130% = 0.0113
so:
[tex]\begin{gathered} y=x+ax \\ so\colon \\ y=x(1+a) \\ _{\text{ }}solve_{\text{ }}for_{\text{ }}x\colon \\ x=\frac{y}{1+a} \\ x=\frac{322507}{0.0113+1} \\ x=\frac{322507}{1.0113} \\ x=318903.3917 \\ x\approx318903.39 \end{gathered}[/tex]Answer:
The original goal was approximately $318903.39
Maybeline is the teacher's assistant today and is correcting homework examples. Help her by selecting correct or incorrect after evaluating each problem.
Answers
Here, we need to remember the signs rules
[tex]\begin{gathered} (+)(+\text{ )=+} \\ (+\text{ )(- )=-} \\ (-\text{ )(+ )=-} \\ (-\text{ )(- )=+} \end{gathered}[/tex]Then
[tex](-4)+(-8)=-4-8=-12[/tex]incorrect.
[tex](-9)-(-8)=-9+8=-1[/tex]correct
[tex](+7)\times(-8)=-56[/tex]correct
[tex](-5)(-2.5)=12.5[/tex]correct
[tex]+\frac{1}{2}\times(+6)=3[/tex]incorrect
For the given functions f and g, find theindicated value.F(x) = x2+ 3x, g(x) =× + 2(f . g) (4)
Answers
Given:
[tex]\begin{gathered} f(x)=\text{ x}^2\text{ + 3x} \\ g(x)\text{ = x + 2} \end{gathered}[/tex]Required:
[tex](f\text{ .g\rparen\lparen4\rparen}[/tex]Recall that:
[tex](f.g)(x)\text{ = f\lparen x\rparen. g\lparen x\rparen}[/tex]Substituting we have:
[tex]\begin{gathered} (f.g)(x)=\text{ \lparen x}^2\text{ + 3x\rparen\lparen x+2\rparen} \\ (f.g)(4)\text{ = \lparen4}^2\text{ + 3\lparen4\rparen\rparen\lparen4 + 2\rparen} \\ =\text{ 28 }\times6\text{ } \\ =\text{ 168} \end{gathered}[/tex]Answer: 168
Find the value of the expression. 07-2 . (131 alw The value is I I
Answers
At a cost of s stickers for c cents, how many stickers can be bought for d dollars
Answers
First, we need to express the amount of dollars d as cents, we can do this as we know that one dollar equals 100 cents, then d in cents would be:
[tex]d(\text{cents)}=d\times100cents[/tex]And from the statement of the question, we know that s stickers cost c cents, we can express that cost per sticker like this:
[tex]\frac{s\text{ stickers}}{c\text{ cents}}[/tex]And if we want to find the amount of stickers that we can buy, we just have to multiply d in cents by the cost per sticker, like this:
[tex]\text{number of sticker we can buy}=\frac{s\times d\times100}{c}[/tex]Hi, im in college and I need help with this here please. Thanks
Answers
The solution of given equations are -4 and 1. The solution of an equation is plotted on the graph.
The given equations are M(d)=2x²+8x-4 and R(d)=2x+4.
What is an equation?In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.
Given, the revenue of each item is same.
That is, M(d)=R(d)
⇒ 2x²+8x-4=2x+4
⇒ 2x²+8x-4-2x-4=0
⇒ 2x²+6x-8=0
⇒ 2x²+8x-2x-8=0
⇒ 2x(x+4)-2(x+4)=0
⇒ (x+4)(2x-2)=0
⇒ x=-4 and x=1
Therefore, the solution of given equations are -4 and 1. The solution of an equation is plotted on the graph.
To learn more about an equation visit:
https://brainly.com/question/14686792.
#SPJ1
I don’t really know if the lines are parallel an explanation would be helpful thanks
Answers
ANSWER
Line K is not parallel to line L.
EXPLANATION
The two angles given are alternate exterior angles. When a line crosses two parallel lines, the alternate exterior angles always sum up to 180 degrees.
So, to confirm if line L is parallel to line K, we check to see if the two given angles sum up to 180 degrees:
[tex]\begin{gathered} 122+68 \\ \Rightarrow190\degree \end{gathered}[/tex]Since they don't sum up to 180 degrees, Line K is not parallel to line L.
One more question please ?
Answers
Solve f(x)= x^4 - 3x^2 + 2 using the radical root theorem and synthetic division.
Answers
Last year, Lisa opened an investment account with $8400. At the end of the year, the amount in the account had decreased by 24.5%. How much is this decrease in dollars? How much money was in her account at the end of last year?
Answers
Answer:
The dec
Explanation:
Given that Lisa opened an investment account with $8400, at the end of the year, the amount in the account had decreased by 24.5%. We want to know how much the decrease is, and how much was in her account at the end of last year.
All we are required to find is what value is 24.5% of $8400
24.5% is the same as:
[tex]\frac{24.5}{100}[/tex]24.5% of $8400 is now:
[tex]\begin{gathered} \frac{24.5}{100}\times8400 \\ \\ =24.5\times84 \\ =2058 \end{gathered}[/tex]Therefore 24.5% of $8400 is $2058
This amount is the decrease.
Finally, at the end of last year, the amount in her account is:
$8400 - $2058 = $6342
Answer: If you do 8400 - 24.5% you get 6342. Then if you do 8400 - 6342 you get 2058. So her account decreased by $2,058 and she had $6342 left in her account at the end of the year.
Savannah invested $5,300 in an account paying an interest rate of 3 5/8 % compounded daily.
Answers
The amount that will be in Savannah's account after 3 years is $6042.
What is compound interest?Compound interest is the interest on savings calculated on both the initial principal and the accumulated interest from previous periods.
To calculate the amount that will be in Savannah's account after 3 years, we use the formula below.
Fromula:
A = P(1+R/365)³⁶⁵ⁿ........... Equation 1Where:
A = Amount P = PrincipleR = Raten = time/yearsFrom the question,
Given:
P = %5300R = 35/8% = 4.375% = 0.04375n = 3 yearsSubstitute these values into equation 1
A = 5300(1+0.04375/365)³ˣ³⁶⁵A = 5300(1.00012)¹⁰⁹⁵A = 5300×1.14A = $6042Hence, there would be $6042 in the account.
Learn more about compound interest here: https://brainly.com/question/24924853
#SPJ1
Compelete question: Savannah invested $5,300 in an account paying an interest rate of 3 5/8 % compounded daily, Assuming no deposits or withdrawals are made, how much money, to the nearest cent, would be in the account after 3 years?
Jan makes four claims about the twopolynomials Any 6x + 1 and 2x. The claims arelisted belowClaim 1 states that when 2x is added to 4xy + 5x +1 the sum is a polynomial.Claim 2 states that when 2x is subtractedfrom xy + 6x + 1 the difference is a polynomial.Claim 3 states that when 4xy + 5x + 1 is multipliedby 2x the product is a polynomial.Claim 4 states that when xy + 6x + 1 is divided by2x the quotient is a polynomialSelect all claims by Jan that are correct.
Answers
A polynomial is defined as an expression which is composed of variables, constants and exponents, that are combined using the mathematical operations such as addition, subtraction, multiplication and division but is never division by a variable.
Therefore claim 4 is false.
When 2x is added, the resultant will be a polynomial.
when 2x is subtracted, the resultant will be a polynomial.
When multiplied by 2x also, the resutant will be a polynomial.
Find the equation of the line that is parallel to Y = x -3 and contains the point (3,-2)
Answers
Given:
The equation of the line is
[tex]y=x-3[/tex]Required:
Find the equation of the line that is parallel to the given line and contains the point (3,-2).
Explanation:
The given equation of the line is
[tex]y=x-3[/tex]Compare the equation with the equation
[tex]y=mx+c[/tex]The slope of the line m = 1.
Since the slope of the parallel lines is equal.
The equation of the line that is parallel to the given line is:
[tex]y=x+b[/tex]This line contains the point (3,-2).
[tex]\begin{gathered} -2=3+b \\ b=-5 \end{gathered}[/tex]Thus the equation of the parallel line is:
[tex]y=x-5[/tex]Final Answer:
[tex][/tex]Determine the slope (m) and y-intercept (b) of the line:y = 2x - 3A) m = 3, b = 2B) m = 2, b = -3C) m = -3, b = 2D) m = 2, b = 3
Answers
slope of line(m) = 2 and intercept on y axis is -2 that is 2 unit in negative y axis.
Equation of line in slope intercept form:
Slope:
Slope is known as tangent of an angle made by positive X-axis.
We know equation of line in slope and intercept form is,
y = mx + b
where,
m is slope of line
b is intercept on y axis
hence for the given equation,
y =2x - 3
slope (m) will be = 2
intercept on y axis (b) = -3
thus,
option (B) will be correct.
To learn more about slope visit:https://brainly.com/question/3605446
#SPJ9
The total bill for repairing Mark’s TV was $211. The repair shop charges $25 an hour for labor plus $16 for parts. How many hours of labor did it take to repair Mark’s TV? Write it in an equation.25/16x = 21125 - 16x = 21125x – 16 =21125x + 16 = 211
Answers
Solution:
Given the total, T is $211;
One hour of labor is $25. So, x hours is $25x
Then, the cost of parts is $16.
Thus;
[tex]25x+16=211[/tex]Determine the length of the longest side of the triangle ABC. Showyour work and round answers to the nearest tenth. *15 in.CB78°10 in.A
Answers
We would make use of the cosine rule,
[tex]\begin{gathered} C^2=A^2+B^2-2AB\cos C \\ C^2=15^2+10^2-2\times10\times15\times\cos 78 \end{gathered}[/tex][tex]\begin{gathered} C^2=325-62.374\text{ =262.626} \\ C=\text{ 16.206 }\approx\text{ 16.2 in} \end{gathered}[/tex]Find each value or measure. Assume that all segments that appear to be tangentare tangent. Find JLK
Answers
Answer
Angle JLK = 31°
Explanation
To answer this, we will use the tangent-chord theorem.
So, the intercepted arc JNL has an angle 298°
Then, we can solve for the tangent chord angle next to it, Angle JLM first by saying
Angle JLM = (Intercepted arc JNL)/2
Angle JLM = (298°/2)
Angle JLM = 149°
Then, we can see that Angle JLM and Angle JLK lie on the same straight line, KLM.
Sum of angles on a straight line is 180°.
Angle JLK + Angle JLM = 180°
Angle JLK + 149° = 180°
Angle JLK = 180° - 149°
Angle JLK = 31°
Hope this Helps!!!
Find the missing length of the triangle. 14 cm 8.4 cm b The missing length is centimeters.
Answers
Answer:
11.2cm
Explanation:
To be able to determine the missing length, we have to apply the Pythagorean Theorem which states that, in a right-angled triangle, the square of the hypotenuse(the longest side) is equal to the sum of squares of the other two sides.
Let's go ahead and find b as follows;
[tex]\begin{gathered} 14^2=8.4^2+b^2 \\ 196=70.56+b^2 \\ 196-70.56=b^2 \\ 125.44=b^2 \\ b=\sqrt[]{125.44}=11.2\operatorname{cm} \end{gathered}[/tex]12. Given that a || b, what is the value of x? (The fiş290ToI41°5
Answers
Ok, so
Here we have the following figure:
We know that both segments are parallel and we want to find the value of x.
For this, remember that the value of x will be the sum of the other two angles:
[tex]\begin{gathered} x=29+41 \\ x=70 \end{gathered}[/tex]This is:
Given the following figure:
The value of x can be find using the following equation:
[tex]x=a+b[/tex]Sanya's car can drive 300 miles in 6 hours. How many miles can she drive in 14 hours?
Answers
Assuming that these variables behave proportionally, we can solve this problem through proportional relationships:
[tex]\begin{gathered} \frac{300}{6}=\frac{x}{14} \\ x=\frac{300\times14}{6} \\ x=\frac{4200}{6} \\ x=700 \end{gathered}[/tex]She can drive 700 miles in 14 hours
14. Sarah draws the following array to solve 49 X 56. What values can be 50 6 determined by this array? 40 9 A 2,000; 240; 450; 54 T B. 2,000; 240; 45; 54 c. 200; 240; 450; 54 ; D. 200; 240; 45; 54
Answers
From the given figure we have 4 different tills
First till has dimensions 50 x 40, then
First till = 50 x 40 = 2000
Second, till has dimensions 6 x 40, then
Second till = 6 x 40 = 240
Third, till has dimensions 9 x 50, then
Third till = 9 x 50 = 450
Fourth till has dimensions 6 x 9, then
Fourth ti;; = 6 x 9 = 54
Then the values that can be determined by the array are
2000, 240, 450, 54
The answer is A
Solve for x.
√x+3 = 2√x-1
Answers
Answer:
x = 16
Step-by-step explanation:
sqrt x + 3 = 2 sqrt x - 1 subtract sqrt x from both sides
3 = sqrt x -1 add 1 to both sides
4 = sqrt x square both sides
16 = x