Answers
Answer:
[tex]\displaystyle \int {\frac{2}{(1 + \sqrt{x})^5}} \, dx = \frac{-(4\sqrt{x} + 1)}{3(\sqrt{x} + 1)^4} + C[/tex]
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
BracketsParenthesisExponentsMultiplicationDivisionAdditionSubtractionLeft to RightAlgebra I
Exponential Rule [Root Rewrite]: [tex]\displaystyle \sqrt[n]{x} = x^{\frac{1}{n}}[/tex]Exponential Rule [Rewrite]: [tex]\displaystyle b^{-m} = \frac{1}{b^m}[/tex]Calculus
Derivatives
Derivative Notation
Basic Power Rule:
f(x) = cxⁿf’(x) = c·nxⁿ⁻¹Antiderivatives - Integrals
Integration Constant C
U-Substitution
U-SolveIntegration Rule [Reverse Power Rule]: [tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]
Integration Property [Multiplied Constant]: [tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]
Integration Property [Addition/Subtraction]: [tex]\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx[/tex]
Step-by-step explanation:
Step 1: Define
[tex]\displaystyle \int {\frac{2}{(1 + \sqrt{x})^5}} \, dx[/tex]
Step 2: Identify Variables
Find the variables u-solve using u-substitution.
U-Substitution
[tex]\displaystyle u = 1 + \sqrt{x}[/tex]
[tex]\displaystyle du = \frac{1}{2\sqrt{x}}dx[/tex]
U-Solve
[tex]\displaystyle x = (u - 1)^2[/tex]
[tex]\displaystyle dx = 2\sqrt{x}du[/tex]
Step 3: Integration
[Integral] Rewrite [Integration Property - Multiplied Constant]: [tex]\displaystyle 2\int {\frac{1}{(1 + \sqrt{x})^5}} \, dx[/tex][Integral] U-Solve: [tex]\displaystyle 2\int {\frac{1}{(1 + \sqrt{(u - 1)^2})^5}} \, 2\sqrt{x}du[/tex][Integral] Simplify: [tex]\displaystyle 2\int {\frac{2\sqrt{x}}{u^5}} \, du[/tex][Integral] Rewrite [Integration Property - Multiplied Constant]: [tex]\displaystyle 4\int {\frac{\sqrt{x}}{u^5}} \, du[/tex][Integral] U-Solve: [tex]\displaystyle 4\int {\frac{\sqrt{(u - 1)^2}}{u^5}} \, du[/tex][Integral] Simplify: [tex]\displaystyle 4\int {\frac{u - 1}{u^5}} \, du[/tex][Integral] Rewrite [Integration Property - Subtraction]: [tex]\displaystyle 4[\int {\frac{u}{u^5}} \, du - \int {\frac{1}{u^5}} \, du][/tex][Integrals] Simplify: [tex]\displaystyle 4[\int {\frac{1}{u^4}} \, du - \int {\frac{1}{u^5}} \, du][/tex][Integrals] Rewrite [Exponential Rule - Rewrite]: [tex]\displaystyle 4[\int {u^{-4}} \, du - \int {u^{-5}} \, du][/tex][Integrals] Reverse Power Rule: [tex]\displaystyle 4[\frac{u^{-3}}{-3} - \frac{u^{-4}}{-4}] + C[/tex]Rewrite [Exponential Rule - Rewrite]: [tex]\displaystyle 4[\frac{-1}{3u^3} + \frac{1}{4u^4}] + C[/tex][Brackets] Multiply: [tex]\displaystyle \frac{-4}{3u^3} + \frac{4}{4u^4} + C[/tex]Back-Substitute: [tex]\displaystyle \frac{-4}{3(1 + \sqrt{x})^3} + \frac{4}{4(1 + \sqrt{x})^4} + C[/tex]Combine: [tex]\displaystyle \frac{-(4\sqrt{x} + 1)}{3(\sqrt{x} + 1)^4} + C[/tex]Topic: AP Calculus AB/BC (Calculus I/II)
Unit: Integration
Book: College Calculus 10e
Related Questions
Let c be a positive number. A differential equation of the form dy/dt=ky^1+c where k is a positive constant, is called a doomsday equation because the exponent in the expression ky^1+c is larger than the exponent 1 for natural growth. An especially prolific breed of rabbits has the growth term My^1.01. If 2 such rabbits breed initially and the warren has 16 rabbits after three months, then when is doomsday?
Answers
Answer:
The doomsday is 146 days
Step-by-step explanation:
Given
[tex]\frac{dy}{dt} = ky^{1 +c}[/tex]
First, we calculate the solution that satisfies the initial solution
Multiply both sides by
[tex]\frac{dt}{y^{1+c}}[/tex]
[tex]\frac{dt}{y^{1+c}} * \frac{dy}{dt} = ky^{1 +c} * \frac{dt}{y^{1+c}}[/tex]
[tex]\frac{dy}{y^{1+c}} = k\ dt[/tex]
Take integral of both sides
[tex]\int \frac{dy}{y^{1+c}} = \int k\ dt[/tex]
[tex]\int y^{-1-c}\ dy = \int k\ dt[/tex]
[tex]\int y^{-1-c}\ dy = k\int\ dt[/tex]
Integrate
[tex]\frac{y^{-1-c+1}}{-1-c+1} = kt+C[/tex]
[tex]-\frac{y^{-c}}{c} = kt+C[/tex]
To find c; let t= 0
[tex]-\frac{y_0^{-c}}{c} = k*0+C[/tex]
[tex]-\frac{y_0^{-c}}{c} = C[/tex]
[tex]C =-\frac{y_0^{-c}}{c}[/tex]
Substitute [tex]C =-\frac{y_0^{-c}}{c}[/tex] in [tex]-\frac{y^{-c}}{c} = kt+C[/tex]
[tex]-\frac{y^{-c}}{c} = kt-\frac{y_0^{-c}}{c}[/tex]
Multiply through by -c
[tex]y^{-c} = -ckt+y_0^{-c}[/tex]
Take exponents of [tex]-c^{-1[/tex]
[tex]y^{-c*-c^{-1}} = [-ckt+y_0^{-c}]^{-c^{-1}[/tex]
[tex]y = [-ckt+y_0^{-c}]^{-c^{-1}[/tex]
[tex]y = [-ckt+y_0^{-c}]^{-\frac{1}{c}}[/tex]
i.e.
[tex]y(t) = [-ckt+y_0^{-c}]^{-\frac{1}{c}}[/tex]
Next:
[tex]t= 3[/tex] i.e. 3 months
[tex]y_0 = 2[/tex] --- initial number of breeds
So, we have:
[tex]y(3) = [-ck * 3+2^{-c}]^{-\frac{1}{c}}[/tex]
-----------------------------------------------------------------------------
We have the growth term to be: [tex]ky^{1.01}[/tex]
This implies that:
[tex]ky^{1.01} = ky^{1+c}[/tex]
By comparison:
[tex]1.01 = 1 + c[/tex]
[tex]c = 1.01 - 1 = 0.01[/tex]
[tex]y(3) = 16[/tex] --- 16 rabbits after 3 months:
-----------------------------------------------------------------------------
[tex]y(3) = [-ck * 3+2^{-c}]^{-\frac{1}{c}}[/tex]
[tex]16 = [-0.01 * 3 * k + 2^{-0.01}]^{\frac{-1}{0.01}}[/tex]
[tex]16 = [-0.03 * k + 2^{-0.01}]^{-100}[/tex]
[tex]16 = [-0.03 k + 0.9931]^{-100}[/tex]
Take -1/100th root of both sides
[tex]16^{-1/100} = -0.03k + 0.9931[/tex]
[tex]0.9727 = -0.03k + 0.9931[/tex]
[tex]0.03k= - 0.9727 + 0.9931[/tex]
[tex]0.03k= 0.0204[/tex]
[tex]k= \frac{0.0204}{0.03}[/tex]
[tex]k= 0.68[/tex]
Recall that:
[tex]-\frac{y^{-c}}{c} = kt+C[/tex]
This implies that:
[tex]\frac{y_0^{-c}}{c} = kT[/tex]
Make T the subject
[tex]T = \frac{y_0^{-c}}{kc}[/tex]
Substitute: [tex]k= 0.68[/tex], [tex]c = 0.01[/tex] and [tex]y_0 = 2[/tex]
[tex]T = \frac{2^{-0.01}}{0.68 * 0.01}[/tex]
[tex]T = \frac{2^{-0.01}}{0.0068}[/tex]
[tex]T = \frac{0.9931}{0.0068}[/tex]
[tex]T = 146.04[/tex]
The doomsday is 146 days
How many solutions does the system of linear equations represented in the graph below have?
No Solution
One Solution
Infinitely many solution
None of the above
Answers
Answer:
No solutions
Step-by-step explanation:
There are no solutions because the two lines have the same gradient (they are parallel)
Have a great day <3 and pls love me
Plz show work thanks
Answers
Answer:
1/sec x- tan x
Step-by-step explanation:
sec x+ tan x
= sec x+ tan x) (sec x - tan x)/(sec x , tan x)
=sec^2x-tan^2x)(sec x- tan x)
(sec^2x-tan^2x=1)
=1/sec x- tan x
A BAKER USES 2/3 CUP OF RYE FLOUR IN EACH LOAF OF BREAD. HOW MANY CUPS OF RYE FLOUR WILL THE BAKER USE IN 9 LOAVES?
Answers
Answer:
6
Step-by-step explanation:
2/3 times 9 = 6
(Also: The Caps really attracted me)
Write each percentage as a fraction in simplest form.
1. 50% 2. 99% 3. 15%
4. 200%
Answers
Answer:
1. 1/2
2. 99/100
3. 3/20
4. 2
Find the area of a room that is 8-feet-wide and 9-feet-long.
Answers
Step-by-step explanation:
Area = l × b
= 9ft × 8ft
= 72ft²
Answer:
72 feet
Step-by-step explanation:
8 (length) • 9 (width) = 72 sq ft
Which expression is equivalent to 64y18 – 1000z6?
Answers
Answer:
8 * (2y⁶ - 5z²) ( 4y¹² +10 y⁶z² + 25z⁴)
Step-by-step explanation:
64y¹⁸ – 1000z⁶ = 8 * (8y¹⁸ - 125z⁶) = 8 * ((2y⁶)³ - (5z²)³)
= 8 * (2y⁶ - 5z²) ( 4y¹² +10 y⁶z² + 25z⁴)
Select all the expressions that represent the large rectangle's total area can't find the answer please help
Answers
A box contains 17 transistors, 5 of which are defective. If are selected at random, find the probability of the statements below.
Answers
Answer:
27% chance that the transistor is defective
5:17
Step-by-step explanation:
The probability that the picked up transistor is defective is 5/17
What is a expression? What is a mathematical equation? What is Probability?A mathematical expression is made up of terms (constants and variables) separated by mathematical operators. A mathematical equation is used to equate two expressions. Equation modelling is the process of writing a mathematical verbal expression in the form of a mathematical expression for correct analysis, observations and results of the given problem. The formula to calculate the probability of occurrence of an event 'A' can be written as -
P(A) = n(A)/n(S)
where -
n(A) = Number of outcomes favorable to event A.
n(S) = Total number of outcomes
Given is that a box contains 17 transistors, 5 of which are defective.
Probability that the picked up transistor is defective is -
P(D) = 5/17
Therefore, the probability that the picked up transistor is defective is 5/17
To solve more questions on Probability, visit the link below -
brainly.com/question/24028840
#SPJ2
HELP ME PLSSSSSSSSSSS IF I FAIL I M Screwed
Answers
Answer:
41
Step-by-step explanation:
what is the answer in fraction form: 2(1/3+1/4)
Answers
Answer:
7/6 or 1*1/6
Step-by-step explanation:
Answer:
the answer is in the picture
I need help
Can you help me please!!!
Answers
Answer:
Step-by-step explanation:
The triangles are similar.
x/(x+4) = 7.2/12
x/(x+4) = 0.6
x = 0.6(x+4)
0.4x + 2.4
x = 6
7.1.14-T Question Help 이 Use the sample data and confidence level given below to complete parts (a) through (d). A drug is used to help prevent blood clots in certain patients. In clinical trials, among 4528 patients treated with the drug, 102 developed the adverse reaction of nausea. Construct a 95% confidence interval for the proportion of adverse reactions. a) Find the best point estimate of the population proportion p. (Round to three decimal places as needed.)
Answers
Answer:
woahhhhhh too much
Step-by-step explanation:
Answer:
7.1.14-T Question Help 이 Use the sample data and confidence level given below to complete parts (a) through (d). A drug is used to help prevent blood clots in certain patients. In clinical trials, among 4528 patients treated with the drug, 102 developed the adverse reaction of nausea. Construct a 95% confidence interval for the proportion of adverse reactions. a) Find the best point estimate of the population proportion p. (Round to three decimal places as needed.)
Find the area of the rectangle
Answers
Answer:
40 mi
Step-by-step explanation:
area= width*length
Answer:
40 mi
Step-by-step explanation:
probally a jok e
[tex]A=l*w[/tex]
[tex]A = 5*8[/tex]
[tex]A=40[/tex]
find the volume of a right circular cone that has a height of 4.2 m and a base with a radius of 3.4 m. Round your answer to the nearest tenth of a cubic meter.
Answers
Answer: 50.8
Step-by-step explanation:
The volume of the given cone is 50.8 m³.
What is cone?A cone is a shape created by connecting all the points of a circular base (which does not contain the apex) to a common point known as the apex or vertex using a series of line segments or lines.
Given that,
Height = h = 4.2 m
Radius of base = r = 3.4 mm
Since we know that,
volume of cone = πr²h/3
now put the values
we get,
volume of cone = π(3.4)²(4.2)/3
= 50.8 square m
Hence,
The volume = 50.8 m³
To learn more about cone visit:
https://brainly.com/question/30253224
#SPJ2
1 point
2) What should come in the place of question mark (?) in the following
letter series based on the English alphabetical order CF HK MP RU? *
O WZ
OEF
GH
OHJ
Answers
Answer:
A) WZ
Step-by-step explanation:
B,C, and D, come before RU
help ill give extra poihts
Answers
Answer:
19 / (19+19+13) = 0.37254901... = ~ 37%
Step-by-step explanation:
19 is self-explanatory
(19+19+13) = 52
19 / 52 = 0.37254901...
0.37254901... = ~37%
brainliest plz
5+10+15+20 in stigma notation
Answers
You mean Sigma notation not stigma ^^
Answer:
Look the attachment
(10 POINTS) answer ASAP if you can
Answers
Answer:
D
Step-by-step explanation:
Because it's 340 meters per second
I only need an equation for this. Plz help me
Answers
m is equal to miles travelled
What does ''a over 3'' or a/3 mean?
Answers
Step-by-step explanation:
It means "a divided by 3". You need to divide a by 3. You can also write it as "1/3a".
Answer:
the answer is above me or below me
What is the area of an equilateral triangle that has the base 12 cm,
and height of 10 cm? *
A 72.4 square centimeters
B 120 square centimeters
C 62.4 square centimeters
D 60 square centimeters
Answers
Answer:
C 62.4square centimetres
Simplify 3y + 6y + 8
Answers
Answer:
Use photo math it will help you.
Answer:
9y+8
Step-by-step explanation:
I hope this helps! You add 3y+6y which is 9 and then you add 8, so the answer is 9y+8
Determine the slope, if it exists, of the graph of the linear equation y=3.1x-3
Answers
This is in y =mx + b
Where m = slope
In one town, there are 240 boys and 260 girls who play
1
sports. Of these, play soccer. How many children
play soccer in all?
Answers
Answer: 500 children play soccer
Step-by-step explanation:
Take the numbers you are given
240 boys
260 girls
now add them together
240 + 260 = 500
Answer:
500
Step-by-step explanation:
There is 240 boys and 260 girls
they all play the same sport
so add does numbers you get 500.
The following expression represents the profit made by John as he mows
lawns: P = 20L - 100
Which description correctly interprets each part of the expression?
Answers
Answer:
P is price per lawn, 20 is the money made per lawn, L is the number of lawns mowed, 100 is his initial cost.
Step-by-step explanation:
Helps please is for nooooowwwww
Answers
Answer:
The correct answer is the tor mar nada tor buda tor buni ami chai
Last one I need help in will give BRAINLY (don’t send me links)
Answers
Answer:
x=11
Step-by-step explanation:
4x+11=5x (angle adb and edf are the same)
11=5x-4x
11=x
x=11
if you want you can check by doing...
(14*11-29)+(5*11)=180
The members of the city cultural center have decided to put on a play once a night for a week. Their auditorium holds people. By selling tickets, the members would like to raise $ every night to cover all expenses. Let d represent the number of adult tickets sold at $. Let s represent the number of student tickets sold at $ each. If all seats are filled for a performance, how many of each type of ticket must have been sold for the members to raise exactly $? At one performance there were times as many student tickets sold as adult tickets. If there were tickets sold at that performance, how much below the goal of $ did ticket sales fall?
Answers
Answer:
a) At $3,300 for 600 seats, the average price per seat is 3300/600 = $5.50. The mix of tickets that results in that average can be found using an X diagram as shown below. The numbers on the right are the differences along the diagonals. When they are multiplied by 2, they add to 600. This shows that the required sales for revenue of $3,300 are
200 adult tickets
400 student tickets
b) When 3 student tickets are sold for each adult ticket, the average seat price is
(3*$4.50 +7.50)/4 = $5.25
Then the shortfall in revenue is ...
$3,300 -480*$5.25 = $780
what is the value of x?
Answers
Answer:
x=37
Step-by-step explanation:
As you can see in the image, both angles are congruent to each other. 74 degrees and 2x degrees. You have to divide 74 by 2 to find what x equals.
74/2=37
x=37