Answer:
Your process is indeed correct!
The full solution is:
[tex]\displaystyle \int\frac{\sin ^3 x}{\sqrt{\cos x}}\, dx= \frac{2}{5} \cos^{{}^{5}\!/\!{}_{2}} x - 2\cos ^{{}^{1}\!/\!{}_{2}}x + C[/tex]
Step-by-step explanation:
We want to evaluate the integral:
[tex]\displaystyle \int \frac{\sin^3(x)}{\sqrt{\cos(x)}}\, dx[/tex]
As you had done, we can rewrite our integral as:
[tex]\displaystyle =\int \frac{\sin(x)(\sin^2(x))}{\sqrt{\cos(x)}}\, dx[/tex]
Using the Pythagorean Identity, this is:
[tex]\displaystyle =\int \frac{\sin(x)(1-\cos^2(x))}{\sqrt{\cos(x)}}\, dx[/tex]
Now, we can make a substitution. Let u = cos(x). Then:
[tex]\displaystyle du = - \sin x \, dx[/tex]
Substitute:
[tex]\displaystyle = \int\frac{1-u^2}{\sqrt{u}} \, \left(- du\right)[/tex]
Simplify:
[tex]\displaystyle = -\int \frac{1}{\sqrt{u}} - \frac{u^2}{\sqrt{u}}\, du[/tex]
Rewrite:
[tex]\displaystyle = -\int u^{{}^{-1}\!/\!{}_{2}} - u^{{}^{3}\!/\!{}_{2}}\, du[/tex]
By the Reverse Power Rule:
[tex]\displaystyle = -\left(2u^{{}^{1}\!/\!{}_{2}} - \frac{2}{5} u^{{}^{5}\!/\!{}_{2}}\right) + C[/tex]
Simplify:
[tex]\displaystyle = \frac{2}{5} u^{{}^{5}\!/\!{}_{2}}-2u^{{}^{1}\!/\!{}_{2}} + C[/tex]
Back-substitute:
[tex]\displaystyle = \frac{2}{5} \cos^{{}^{5}\!/\!{}_{2}} x - 2\cos ^{{}^{1}\!/\!{}_{2}}x + C[/tex]
Answer:
= - 2 [tex]\sqrt{cos(x)}[/tex] + 2 cos⁵/₂ (x) + C
5
Step-by-step explanation:
∫ sin³ (x) dx
[tex]\sqrt{cos(x)}[/tex]
= ∫ sin² (x) sin (x) dx
[tex]\sqrt{cos(x)}[/tex]
= ∫ (1 - cos² (x) sin (x) dx
[tex]\sqrt{cos(x)}[/tex]
= ∫ - 1 - u² du
√u
= ∫ - 1 + u³/₂ du
√u
= - ∫ 1 du + ∫ u³/₂ du
√u
substitute it back
= - 2 √u + 2 cos⁵/₂ (x)
5
add constant, therefore
= - 2 [tex]\sqrt{cos(x)}[/tex] + 2 cos⁵/₂ (x) + C
5
Show the steps to get full credit
3 (2x + 2) = 18 + 4(x - 5)
Answer:
[tex]x=-4[/tex]
Step-by-step explanation:
So we have the equation:
[tex]3(2x+2)=18+4(x-5)[/tex]
Distribute the left and the right:
[tex]3(2x)+3(2)=18+4(x)+4(-5)[/tex]
Simplify:
[tex]6x+6=18+4x-20[/tex]
Combine on the right:
[tex]6x+6=4x-2[/tex]
Subtract 4x from both sides. The right cancels:
[tex](6x+6)-4x=(4x-2)-4x\\2x+6=-2[/tex]
Subtract 6 from both sides. The left cancels:
[tex](2x+6)-6=(-2)-6\\2x=-8[/tex]
Divide both sides by 2:
[tex](2x)/2=(-8)/2\\x=-4[/tex]
Thus, the value of x is -4.
Answer:
x = -4
Step-by-step explanation:
3(2x + 2)=18 + 4(x - 5)
6x + 6= 18 +4x - 20
6x - 4x = -2-6
2x = -8
x = -4
PLEASE HELP ME HELP MY 7TH GRADER! The ratio of picnic tables to garbage cans in a new national park should be 8:3. The park design shows plans for picnic tables in a small campground? There are 40 small picnic tables in the campgrounds and 120 large picnic tables at the camp grounds.
Answer:
small: 40:15 large: 120:45 all: 160:60
Step-by-step explanation:
the way i did this was
40÷8=5 i did this because we now need to multiply 5×3=15 because thats how we get the number of trash cans.
now do the same thing for the other one
120÷8=15 → 15×3=45
an easy way to check ur answers is to divide the ratio that is given to you (8:3 so 8÷3) and see if it gets the same number as ur new ratio (40:15 so 40÷15)
Im in 8th grade and I hoped this helped :)
14. A rare bacterial culture is being grown in a lab. As the days progress, the cells multiply and grow.
After 1 day, there is only 1 cell.
After 2 days, there are 9 cells.
After 3 days, there are 20 cells.
After 4 days, there are 34 cells.
Awhich recursive equation represents the pattern? (2 points)
an = an - 1 + 3n
an = an - 1 +n + 2)
an = an - 1 + 3(n + 2)
an = an - 1 + (3n + 2)
Answer:
a[n] = a[n-1] +(3n +2)
Step-by-step explanation:
The difference between Day 1 and Day 2 is 9 - 1 = 8 cells. The difference between a2 and a1 in each of the answer choices is a2 -a1 = ...
3n = 3(2) = 6 ≠ 8
n+2 = 2+2 = 4 ≠ 8
3(n+2) = 3(2+2) = 12 ≠ 8
3n +2 = 3(2) +2 = 8 . . . . . . this is the appropriate choice:
[tex]\boxed{a_n=a_{n-1}+(3n+2)}[/tex]
What is the volume of the rectangular prism (please help urgent)
Answer:
The answer is 1 1/9 ft
Step-by-step explanation:
Multiply 5/3 x 2/3 x 1
Write the following phrase as a variable expression. Use x to represent “a number” The sum of a number and fourteen The translation is...
Answer:
x+14.
Step-by-step explanation:
Sum of a number and 14 is x+14.
Heyy really need help on this question anyone would Iike to help ?
Given: F(x) = 2x - 1; G(x) = 3x + 2; H(x) = x 2 Find F{G[H(2)]}. 27 71 121
Answer:
Step-by-step explanation:
h(2)= 2
g(2)= 3(2) + 2 = 6 + 2 = 8
f(8)= 2(8) - 1 = 16 - 1 = 15
Answer:
27
Step-by-step explanation:
Set up the composite result function.
F(G(H(x)))
Evaluate F(G(H(x))) by substituting in the value of H into G.
Multiply 3 by x^2
F(G(x2))=F(3x2+2)
Evaluate F(G(x)) by substituting in the value of G into F
F(3x2+2)=2(3x2+2)-1
Simplify each term.
Apply the distributive property.
F(3x2+2)=2(3x2)+2⋅2-1
Multiply 3 by 2
F(3x2+2)=6x2+2⋅2-1
Multiply 2 by 2
F(3x2+2)=6x2+4-1
Subtract 1 from 4
F(3x2+2)=6x2+3
Evaluate the result function by replacing x with 2 in the function.
6(2)2+3
Simplify the evaluated function.
Simplify each term.
Raise 2 to the power of 2
6⋅4+3
Multiply 6 by 4
24+3
27
3x + 5 = 2x + 15 solve for x
3x +5 = 2x +15
Subtract 5 from both sides:
3x = 2x + 10
Subtract 2x from both sides:
X = 10
Answer:
The answer is x=10.
3x-2x=15-5.
x=10.
Find the distance between the two points rounding to the nearest tenth (if necessary).
(-3,-3) and (-6,1)
Answer:
5Step-by-step explanation:
[tex](-3,-3) =(x_1,y_1)\\ (-6,1)=(x_2,y_2)\\\\d = \sqrt{(x_2-x_1)^2 +(y_2-y_1)^2}\\ \\d = \sqrt{(-6-(-3))^2+(1-(-3))^2}\\ \\d = \sqrt{(-6+3)^2+(1+3)^2}\\ \\d = \sqrt{(-3)^2 +(4)^2}\\ \\d = \sqrt{9+16}\\ \\d = \sqrt{25}\\ \\d =5[/tex]
elena's retirement part will cost $16 if she invites 8 guests. If there are 10 guest how much will elena retirement party cost?
Answer:
$20
Step-by-step explanation:
Please mark me as brainliest
Answer:
Elena's retirement party will cost $20 if she invites 10 guests.
Step-by-step explanation:
We know this because the problem tells us that the party will cost $16 if she invites 8 guests, which means that every guest costs $2 and if we multiply 2 and 10 we get 20.
Which of the following statements is true about completing a square? A. When completing the square, subtract the square of half the coefficient of x from both sides of the equation. B. When completing the square, add the square of half the coefficient of x to only the x^2+bx side of the equation. C. When completing the square, add half the coefficient of x to both sides of the equation. D. When completing the square, add the square of half the coefficient of x to both sides of the equation.
Answer:
D
Step-by-step explanation:
So, let's say that we have the following equation:
[tex]x^2+bx=0[/tex]
And we want to complete the square.
What we need to do is to divide the coefficient of x by half. Square that. Then add that number to both sides.
The answer that reflects this is choice D.
For instance if we have:
[tex]x^2+4x=0[/tex]
Divide 4 by 2 and then square it.
4/2 is 2. 2²=4. Add 4 to both sides. Thus:
[tex](x^2+4x)+4=0+4[/tex]
And now, we can complete the square:
[tex](x+2)^2=4[/tex]
Can you please help me I will give out extra points.
negatives of a social structure
What the measure of AOB?
Answer:
∠ AOB = 17.5°
Step-by-step explanation:
∠ AOB + 110 + ∠ COD = 180 ( sum of angles in straight line )
[tex]\frac{1}{3}[/tex] ∠ COD + 110 + ∠ COD = 180 ( subtract 110 from both sides )
[tex]\frac{4}{3}[/tex] ∠ COD = 70 ( multiply both sides by 3 to clear the fraction )
4 ∠ COD = 210 ( divide both sides by 4 )
∠ COD = 52.5°
Thus
∠ AOB = [tex]\frac{1}{3}[/tex] × 52.5° = 17.5°
A dog walker charges a flat rate of $6 per walk plus an hourly rate of $30. How much does the dog walker charge for a 45 minute walk? Write an equation in function notation for the situation, and then use it to solve the problem. Determine if the given statement is True or False.
The dependent variable is the number of hours.
A True B False
The function for the walker's fee is f(h) = 30h + 6.
A True B False
The dog walker charges $22.50 for a 45 minute walk.
A True B False
Answer:
- False
- True
- false
Step-by-step explanation:
- The variable that changes is the price
- 30 is the slope and 6 is the y-int
- 30(.75) + 6 = 28.5
4) A bank is offering 3.5% simple interest on a savings account. If you deposit
$12,000, how much interest will you earn in two years? *
A $420
B $840
C $4,200
O D $8,400
Answer:
B. $840
Step-by-step explanation:
Given that
Rate of simple interest = 3.5%
Principal amount = $12,000
Time = 2 year
To find:
Interest earned = ?
Solution:
Formula for Simple Interest is given as:
[tex]S.I. = \dfrac{PRT}{100}[/tex]
Where P is the principal amount
R is the annual Rate of interest
T is the time for which principal was invested.
Here, we are given that:
P = $12,000
R = 3.5%
T = 2 years
Putting all the values in the formula:
[tex]SI= \dfrac{12000\times 3.5 \times 2}{100}\\\Rightarrow SI= 12\times 35 \times 2\\\Rightarrow SI = \$840[/tex]
So, answer is:
B. $840
How to find the area of a triangle
Answer:
A = 1/2bh
Step-by-step explanation:
The formula for an area of a triangle is always going to be one-half of the base times the height.
A storage pod has a rectangular floor that measures 19 feet by 13 feet and a flat ceiling that is 5 feet above the floor. Find the area of the floor and the volume of the pod.
Answer:
Area= 247 feet²
Volume= 1235 feet³
Step-by-step explanation:
A storage pod has a rectangular floor that measures 19 feet by 13 feet and a flat ceiling that is 5 feet above the floor.
Area of the floor= length*width
Length= 19 feet
Width= 13 feet
Area= 19*13
Area= 247 feet²
Volume of the pod= area*height
Height of the pod= 5 feet
Area= 247 feet²
volume= 247*5
Volume= 1235 feet³
P(A) = 0.33, P(A and B) = 0.25, find P(B
Answer:
≈0.76
Step-by-step explanation:
P(A) = 0.33
P(A and B) = 0.25
P(B) = ?
-------------
Probability of two events happening is the product of individual probabilities
P(A and B) = P(A)×P(B)Then to find the probability of one of the events:
P(B) = P(A and B) / P(A)P(B) = 0.25 / 0.33 ≈ 0.761. Original Sentence: My friend Cody can eat a sandwich faster than any human or animal I have ever seen. New Word: Definition: Two synonyms also considered:
Answer:
My friend Cody can eat a sandwich swiftlier than any human or animal I have ever seen.
a polynomial has one quadratic factor and 3 linear factors. One of the linear factors has multiplicity two. What is the degree of the polynomial
Answer:
5
Step-by-step explanation:
a polynomial has one quadratic factor and 3 linear factors. One of the linear factors has multiplicity two. What is the degree of the polynomial
A polynomial with one quadratic obtains the forms ( ax² +bx +c ) with 3 linear factors.
Suppose the three linear fractions are :
(x- P) (x-Q) (x- R)
∴
The polynomial = ( ax² +bx +c )(x- P) (x-Q) (x- R)
By factorization, the highest degree of the polynomial = 5
Evaluate f(x) = 5x - 1 given the inputs (-1, 0,1,2)
Answer:
-6, -1, 4, 9
Step-by-step explanation:
You can substitute the values and do the arithmetic:
f(-1) = 5(-1) -1 = -5 -1 = -6
Or, you can recognize that the output values will increase by 5 for each increase of 1 in the input value.
f(0) = -6 +5 = -1
f(1) = -1 +5 = 4
f(2) = 4 +5 = 9
The (input, output) pairs are ...
(-1, -6), (0, -1), (1, 4), (2, 9)
What is the sum of 73.04 × 1.2
Answer:
87.648
Step-by-step explanation:
It may be hard to multiply decimals. However, if you can break the decimal up into two separate equations, it can make it easier. Solving 73.04 x 1.2 is the same as solving (73.04 x 1) + (73.04 x 0.2). You already can say that 73.04 x 1 is 73.04. And then you can solve 73.04 x 0.2. Solving 73.04 x 0.2 is the same as solving 73.04 divided by 5, which will get you 14.608. Finally, all you have to do is add the two sums (73.04 + 14.608) which will get you your answer of 87.648.
Help Me Please!!!! I don’t get it
Answer/Step-by-step explanation:
In mathematics, there's an order to follow in solving calculations like the problems given above. The acronym, PEDMAS illustrates or makes us remember which order to follow. The acronym stands for:
P - Parenthesis. Given any problem to solve, first, solve any operation within a parenthesis. That is, expressions that are grouped together.
E - Exponents. Next in line of priority order is to solve expressions involving exponents.
D - Division
M - Multiplication
A - Addition, and
S - Subtraction.
Now, solve each expression given in the correct order of operation using PEDMAS:
10 - 3³ ÷ 9
10 - 27 ÷ 9
10 - 3
= 7 (the order of operation is EDS = Exponent, Division, and lastly, Subtraction)
7² ÷ (4 + 3)
7² ÷ 7
49 ÷ 7
= 7 (order of operation = PED)
7 × 5 - 2²
7 × 5 - 4
35 - 4
= 31 (EMS)
(6 + 2²) × 10
(6 + 4) × 10
10 × 10
= 100 (PM)
3 × 6 + 8²
3 × 6 + 64
18 + 64
= 82 (EMA)
4³ - 10 ÷ 5
64 - 10 ÷ 5
64 - 2
= 32 (EDS)
3² × 2 - 9
9 × 2 - 9
18 - 9
= 9 (EMS)
9 × 3² - 8
9 × 9 - 8
81 - 8
= 73 (EMS)
6² ÷ 3 - 5
36 ÷ 3 - 5
12 - 5
= 7 (EDS)
(9 - 5)² ÷ 4
(4)² ÷ 4
16 ÷ 4
= 4 (PED)
Solve the following absolute value equation. If needed, write your answer as a fraction reduced to lowest terms. Please explain your answer |y−43|−|y−35| =0
Answer:
y = 39
Step-by-step explanation:
|y−43|−|y−35| =0
add |y−35| to each side
|y−43|= |y−35|
This tells us that y must be between 35 and 43
That means |y−35| is positive so we can remove the absolute value signs
|y−43| is negative so we negate each term ( or multiply by -1)
-y+43 = y - 35
Add y to each side
y-y+43 = y+y-35
43 = 2y -35
Add 35 to each side
43+35 = 2y
78 = 2y
Divide by 2
78/2 = 2y/2
39 = y
Answer:
Step-by-step explanation:
Three students need to produce a prime factorization of 48. Donna states that the first factors in the tree should be 6 and 8. Larry states that the first factors in the tree should be 4 and 12. Trish states that the initial factors of 48 do not affect the prime factorization. Explain why Trish is correct. Why is trish correct?
Answer:
Below.
Step-by-step explanation:
The initial factors do not affect the prime factorisation.
Continuing the factorisation will lead to the same result whichever initial factors you choose.
48
4 * 12
2*2 * 2*2*3
= 2*2*2*2*3
48
6 * 8
2*3 * 2*2*2
= 2*2*2*2*3.
Answer:
A prime factorization is the product of the prime numbers that multiply to give the original number. 48 will have a prime factorization of 2(2)(2)(2)(3), or 24(3), regardless of the initial factors.
Step-by-step explanation:
It is correct!
If you were to describe a rotation of triangle ABC what information would you need to include in your description
You would need to include the center of rotation. For many problems, the center is usually the origin (0,0). This is a default assumed center. Though the center could easily be any other point.
In addition, you also need the amount of rotation that is being applied (eg: 90 degrees).
The last bit of info you need is whether you're going clockwise or counterclockwise. For 180 degree rotations, it doesn't matter, but for everything else it does matter.
a saleswoman earns 60% commission on all that she sells. Last month she sold $400 worth of merchandise. How much commission (In dollars) did she earn last month?
Answer:
$240
Step-by-step explanation:
If she earns 60% commission, that means she only earns 60% of all sales. So if you take that 60% or (.60) you can multiply that 400 by .60, that equates to 240.
The sum of two numbers is fifteen. One number is three less than the other. Determine the system of equations.
Answer:
let, one number be x
then, other number be x-3
X+X-3=15
or, 2X=15+3
or,2X=18
or,X=18÷2
or,X=9
one number =9
then another number=X-3=9-3=6
The system of equations with two numbers x ad y are,
⇒ [tex]x + y = 15[/tex]
⇒ [tex]x = y - 3[/tex]
Given that,
The sum of two numbers is fifteen.
And, One number is three less than the other.
Let us assume that,
Two variables for the numbers in the system of equations are x and y.
Hence, we can write the system of equations as,
The sum of two numbers is fifteen.
⇒ [tex]x + y = 15[/tex]
And, One number is three less than the other.
⇒ [tex]x = y - 3[/tex]
So, the system of equations is,
⇒ [tex]x + y = 15[/tex]
⇒ [tex]x = y - 3[/tex]
Learn more about systems of equations at:
brainly.com/question/14323743
#SPJ4
NEED HELP ASAP
Which angle pairs are supplementary? Check all that
apply.
<1 and <2
<4 and <3
<4 and <5
<7 and <5
<3 and <6
Answer:
∠4 and ∠3
∠4 and ∠5
∠3 and ∠6
Step-by-step explanation:
A pair of angles is said to be supplementary when the 2 angles add up to give 180°.
Two right angles add up to give 180°. Also, if we have two angles on a straight line, their sum = 180°, according to the linear pair property.
Thus, since m< 3 is given as 90°, m<4 = 90°.
m<4 + m<3 = 180°.
Therefore, <4 and <3 are supplementary.
<4 and <5, <3 and <6 are both linear pairs, therefore they are also supplementary.