Recall some identities:
sin²(x) + cos²(x) = 1
cos(2x) = cos²(x) - sin²(x)
and from the first identity, you can establish that
cos(2x) = 2 cos²(x) - 1
So, rewrite the integrand as much as possible in terms of cos(x) :
[tex]\sin^3(x) \cos(2x) = \sin(x) \sin^2(x) (\cos^2(x)-\sin^2(x)) \\\\ \sin^3(x) \cos(2x) = \sin(x) (1-\cos^2(x)) (2\cos^2(x) - 1)[/tex]
Then in the integral, substitute u = cos(x) and du = -sin(x) dx. With some rewriting, you end up with
[tex]\displaystyle \int_1^2 \sin^3(x) \cos(2x) \,\mathrm dx = -\int_{x=1}^{x=2} (1-\cos^2(x))(2\cos^2(x)-1) (-\sin(x)\,\mathrm dx) \\\\ = -\int_{u=\cos(1)}^{u=\cos(2)} (1-u^2)(2u^2-1)\,\mathrm du \\\\ = \int_{\cos(1)}^{\cos(2)} (1-3u^2+2u^4) \,\mathrm du \\\\ = \left(u-u^3+\frac25u^5\right)\bigg|_{\cos(1)}^{\cos(2)} \\\\ = \left(\cos(2)-\cos^3(2)+\frac25\cos^5(2)\right) - \left(\cos(1)-\cos^3(1)+\frac25\cos^5(1)\right) \\\\ = \boxed{\cos(2) - \cos^3(2) + \frac 25 \cos^5(2) - \cos(1) + \cos^3(1) - \frac25 \cos^5(1)}[/tex]
Choose the repeating decimal that is equal to
the fraction on the left.
0.17
0.351
0.17
0.351
0.351
58
165
79
225
13
37
ОО
ОООО
8
45
ООО
17
99
Step-by-step explanation:
[tex] \frac{58}{165}[/tex]
[tex] \frac{79}{225} [/tex]
[tex] \frac{13}{37} [/tex]
[tex] \frac{8}{45} [/tex]
[tex] \frac{17}{99} [/tex]
The answer is in the picture shown here
Cecelia receives two credit card offers in the mail. Each has a 0% introductory APR. After the introductory period, the first card has an APR of 19.99% and the second card has an APR of 24.99%. If each card has a late fee of $35 and an annual membership fee of $25, which card should she choose and why?
Answer:
1. We are told they have similar APR (starting) and Annual fee of 25 and similar late fee of 35
Then we must consider the APR after the initial value 19.999% < 24.99%Hence she should price the first card with a 19.99% APR
How high is a tree that cast a26ft shadow at the same time a 6ft post casts a shadow which is 11ft long
Answer:
[tex]14\frac{2}{11}[/tex] ft
Step-by-step explanation:
Let the height of the tree equal x. Shadows are proportional to the height of an object, so we can write a proportion where to numerators are the object and the denominators are the shadow:
[tex]\frac{x}{26} =\frac{6}{11}[/tex]
Now, to solve for x, we can cross-multiply:
11x = 6*26
11x = 156
x = [tex]14\frac{2}{11}[/tex]
What is the value of (-1/4 - 1/2) ÷ (-4/7)
Answer:
1 5/16
Step-by-step explanation:
(-1/4 - 1/2) ÷ (-4/7)
Parentheses first
Get a common denominator
(-1/4 - 1/2 *2/2)
(-1/4 - 2/4)
-3/4
Replace in the equation
(-3/4 ) ÷ (-4/7)
Copy dot flip
-3/4 * -7/4
21/16
1 5/16
[tex]\\ \rm\Rrightarrow \left(\dfrac{-1}{4}-\dfrac{1}{2}\right)\div \dfrac{-4}{7}[/tex]
[tex]\\ \rm\Rrightarrow \dfrac{-1-2}{4}\times\dfrac{-7}{4}[/tex]
[tex]\\ \rm\Rrightarrow \dfrac{-3}{4}\times \dfrac{-7}{4}[/tex]
[tex]\\ \rm\Rrightarrow \dfrac{21}{16}[/tex]
[tex]\\ \rm\Rrightarrow 1\dfrac{5}{16}[/tex]
Plz help me with these with steps
Answer: There are 100 blue marbles.
Step-by-step explanation:
252 x 2/7 = 72
72 is the number of the red marbles.
(252-72) x 4/9 = 80
80 is the number of the green marbles.
252-72-80 = 100
There are 100 blue marbles.
There is another bag of red and yellow footballs with a ratio of 4:3. If there are 20 red footballs in the bag, how many footballs are there in total?
Answer:
35
Step-by-step explanation:
take no. of red footballs be 4x and yellow footballs be 3x
4x=20, x=5
=> 3(5)= 15
total no. of footballs= 20+15= 35
Is 9pi rational or irrational
Answer:
irrational
Step-by-step explanation:
every number that times with pi are irrational, only zero gives a rational number
A glucose molecule is the
smallest type of protein
monomer of starch and cellulose
largest of the carbohydrate polymers
fundamental unit of genetic material
A glucose molecule is the smallest type of protein monomer of starch and cellulose largest of the carbohydrate polymers fundamental unit of genetic material.
A glucose molecule is the monomer of starch and cellulose.
Step-by-step explanation:
Glucose is an organic molecule composed of carbon atoms, oxygen atoms, and hydrogen atoms.It has a molecular formula of [tex]C_6H_{12}O_6[/tex]It is a monosaccharide sugar, carbohydrateAn important source of an energy in living organisms.Glucose gives two different natural polymers named: starch and cellulose. This polymer is formed by linking small units of glucose molecules with each other. And this small unit which repeats in the structure of its polymer is known as a monomer.Glucose in the monomer in its natural polymer that was starch and cellulose.So from this, we can conclude that A glucose molecule is the monomer of starch and cellulose.
Learn more about glucose here:
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Solve for w.
|5w + 7) = |5w + 8)
please help :)
Answer:
w=3/2
Step-by-step explanation:
[tex]|5w+7|=|5w+8|\\5w + 7 =\pm (5w+8)\\5w + 7 = 5w + 8 \text{ or } 5w + 7 = -5w-8\\7 = 8 (\text{abusrd}) \text{ or } 10 w =15\\w= 15/10 = 3/2.[/tex]
draw a number line and represent the following rational number 5 upon 4
plz help me in this question
class 7
i will mark brainliest
Answer:
First we need to convert this fraction number to decimal then only we caan mark that on a number line
[tex]\frac{5}{4}[/tex] divide
You will get 1.25
Now mark 1.25 on number line
jasper had 3 5/6 feet of rope. He cut off two pieces that were 1 2/3 feet each for a project. How many rope does Jasper have left?
Answer:
Jasper have left 1/2 feet
Step-by-step explanation:
3 5/6 - (1 2/3 x 2)
= 23/6 - 10/3
= 23/6 - 20/6
= 1/2
Answer:
Jasper has [tex]3/6[/tex] feet of rope left.
Step-by-step explanation:
[tex]1 \frac{2}{3} + 1\frac{2}{3} = 3 \frac{1}{2} = 3\frac{2}{6} \\\\3 \frac{5}{6}- 3\frac{2}{6}= \frac{3}{6}[/tex]
A good doctor recommends that a person who weighs 65 pounds be given 200
milligrams of medicine. If the dosage were proportional to a person's weight, how
many milligrams of medicine would the doctor recommend for a person who
weighs 96 pounds?
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Answer:
295 mg
Step-by-step explanation:
The proportion can be written ...
dose/(96 lb) = (200 mg)/(65 lb)
dose = (96/65)(200 mg) ≈ 295.4 mg . . . . . multiply by 96 lb, cancel lb units
The doctor might recommend 295 mg for a 96-pound patient.
Average rate of change question
Answer:
Step-by-step explanation:
which graph shows x =4?
Answer:
second graph shows x=4 because you can see from origin point and counting will makex=4
Mark bought 8 boxes a week later half of all his boxes were destroyed in a fire there are now only 20 boxes left with how many did he start?
Let it b represent how many boxes he started with
Equation or inequality:
Mark began with:
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Answer:
equation: 1/2(s +8) = 20Mark began with 32 boxesStep-by-step explanation:
Let s represent the number Mark started with. After buying 8 more, he then had s+8. After the fire, he had ...
(1/2)(s +8) = 20
This equation can be solved by multiplying by 2, then subtracting 8.
s +8 = 40
s = 32
Mark began with 32 boxes.
Calculate simple interest and compound interest assuming that principal amount is $ 10,000; interest rate is 9% for three years. What is the amount different between compound and simple interest?
Step-by-step explanation:
Please mark me as brainlyest
simplify -(2a -2) -(a + b)+(a-6+b)
Step-by-step explanation:
-2a-2-a-b+a-6+b
-2a-a+a-b+b-2-6
-2a+0-8
-2a-8
Answer:
-2a-8
Step-by-step explanation:
If you like my answer than please mark me brainliest thanks
HELP MEE !!!!!!!!!!!!!!!!!!!!
Answer:
It is D
9 * (12 + 7) = 9 * 19 = 171
now we test to see if 9 * 12 + 9 * 7 is the same
9 * 12 = 108
9 + 7 = 63
108 + 63 = 171
Step-by-step explanation:
Harvey is 3 times as old as Jane. The sum of their ages is 48 years. Find the age of each.
Jane is
years old.
Harvey is
years old.
Answer:
Jane is 12 and Harvey is 36
Step-by-step explanation:
3x+x=48
4x=48
x=12
3(12)=36
1(12)=12
in the number 58, the value of the digit 8 is ----
and the value of the digit 5 is __ ?
Answer:
8 is one's. 5 is tens.
Step-by-step explanation:
the 5 is in the tens place, and the 8 is in the ones place.
A student rotated figure A 270° clockwise about the origin to create figure D. Evaluate whether the student performed the rotation correctly.
Step-by-step explanation:
The student incorrectly performed a rotation of 90° clockwise.
Answer:
C, the student incorrectly preformed a rotation of 90 degrees clockwise.
Step-by-step explanation:
What's the X-Intercepts of (2x-5)(x+1)=9 or (2x-5)(x+1)-9
Answer:
Step-by-step explanation:
Either way works. It comes to the same thing.
(2x-5)(x+1)=9
F: 2x*x = 2x^2
O: 2x*1 = 2x
I: -5 * x = -5x
L: -5*1 = -5
2x^2 + 2x - 5x - 5 = 9 Subtract 9 from both sides
2x^2 - 3x - 14 =0 Divide by 2 (it makes life easier).
Factoring you get
(2x - 7)(x + 2)
2x - 7 = 0
2x = 7
x = 3.5
x + 2 = 0
x = -2
The x intercepts = (3.5,0) and (-2,0)
I've put a graph up for the given equation.
Don Simón repartió 60 canicas entre sus dos
hijos, de modo que el mayor recibió tres veces lo
que el menor. ¿Cuántas canicas recibió el hijo
mayor?
Un sistema de ecuaciones es un conjunto de ecuaciones que operan con el mismo conjunto de variables, de tal forma que buscamos solucionar ambas ecuaciones al mismo tiempo.
En este problema querremos encontrar un sistema de ecuaciones para poder encontrar la cantidad de canicas que recibe el hijo mayor.
Encontraremos que recibe 45 canicas
La información que se nos da es:
"Don simón repartió 60 canicas entre ambos hijos"
"El hijo mayor recibe 3 veces la que el menor"
Definamos las variables:
x = cantidad que recibe el hijo mayor
y = cantidad que recibe el hijo menor.
Como sabemos que la cantidad total de canicas es 60, entonces tenemos:
x + y = 60
Como tambien sabemos que el mayor recibe 3 veces lo que el menor, podemos escribir:
x = 3*y
Entonces tenemos 2 equaciones, las cuales podemos combinar en un sistema de ecuaciones.
x + y = 60
x = 3*y
Para resolver este sistema, el primer paso sería aislar una de las variables en una de las ecuaciones. Como queremos resolverlo para x, debemos aislar la otra variable.
Podemos hacerlo en la segunda ecuación:
x/3 = y
Ahora podemos reemplazar esto en la primer ecuación:
x + y = 60
x + x/3 = 60
(4/3)*x = 60
x = (3/4)*60 = 45
x = 45
Podemos concluir que el mayor recibe 45 canicas.
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Please help I will mark brai list if u are right
Answer:
Umm I'm only in middle school umm I think it's A bc there pointing in the same line
<7 and <8 are complimentary angles. <5 = <8 and m<6 = 29. Find the measure of angle 7 and angle 5
Here <5=<8 then
<7=<6=29[tex]\\ \sf\longmapsto <7+<8=90[/tex]
[tex]\\ \sf\longmapsto 29+<8=90[/tex]
[tex]\\ \sf\longmapsto <8=90-29[/tex]
[tex]\\ \sf\longmapsto <8=61[/tex]
<5=61°evaluate the expression
Step-by-step explanation:
13
[tex]3 + (20 \div 5) - ( {2}^{2}) [/tex]
[tex]3 + 4 - 4 = 3[/tex]
15
[tex] \frac{10 - {3}^{2} }{20 - (3 \times 4)} [/tex]
[tex] \frac{10 - 9}{20 - 12} [/tex]
[tex] \frac{1}{8} [/tex]
need help with this!
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Answer:
a(n) = 1/2(-8)^(n-1)
Step-by-step explanation:
The general term of a geometric sequence is ...
a(n) = a(1)×r^(n-1)
where a(1) is the first term, and r is the common ratio.
Here, the first term is a(1) = 1/2. The common ratio is the ratio of any two sequential terms: r = 32/-4 = -8. Then the explicit formula is ...
a(n) = 1/2(-8)^(n-1)
The points (a,b) and (c,d) form a segment, and the points (d,e) and (d,f) form a segment. Create an equation assuming the segments are congruent
If the segments are congurent they are of equal length.
So constructing an equation will not be hard.
Take for example [tex]y=x[/tex] and limit x to be between and including 0 and 1 for the first segment. In this case our segment is [tex](a,b)\to(c,d)=(0,0)\to(1,1)[/tex].
Now, d has been fixated to the value of 1 and we need to construct a segment from [tex](1,e)\to(1,f)[/tex].
Since both x coordinates of the endpoints of the segment are fixed to be 1, we cannot run anymore, that is, we fixated our run. But on the rise (y-axis direction) we can still move one unit up.
Let e be 1 and f be 2. The distance between [tex](1,1)[/tex] and [tex](1,2)[/tex] is 1 which is also the distance between [tex](0,0)[/tex] and [tex](1,1)[/tex].
Now we are asked to find the equation of both segments.
First segment is described by [tex]y=x[/tex] with limited domain of [tex]0\leq x\leq1[/tex].
Second segment is described by [tex]x=2[/tex] with again limited domain of [tex]1\leq y\leq2[/tex].
Hope this helps :)
The length, l of a segment with points (x₁, y₁), and (x₂, y₂) is presented as follows; [tex]l = \sqrt{\left (y_{2}-y_{1} \right )^{2}+\left (x_{2}-x_{1} \right )^{2}}[/tex]
An equation that can be created given that the segment with points (a, b) and (c, d), and the segment with points (d, e), and (d, f) are congruent is presented as follows;
√((d - b)² + (c - a)²) = f - e
The reason the above equation is correct is presented as follows;
Given that the segments are congruent, therefore;
Length of (a, b) and (c, d) = Length of (d, e) and (d, f)
Length of (a, b) and (c, d) = √((d - b)² + (c - a)²)
Length of (d, e) and (d, f) = √((f - e)² + (d - d)²) = f - e
Which gives the equation;
√((d - b)² + (c - a)²) = f - e
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There are 25 students' names in a hat. You choose 5 names. Three are
boys' names and two are girls' names. How many of the 25 names would
you expect to be boys' names?
Show all of your work do in 10 minutes
Answer:
15
Step-by-step explanation:
You have to calculate the boys name using ratio expression.
3/5 × 25
= 15
Help plsssssssssssssssssssssss
Answer:
Step-by-step explanation:
let Garrett orders x stickersfrom each company.
then
30+0.80x=14+1.20x