Answer:
[tex]\int\limits {cos(x)\ sin(2x)\ sin(x)} \, dx = \frac{4x-sin(4x)}{16} +c[/tex]
Step-by-step explanation:
Given
[tex]\int\limits {cos(x)\ sin(2x)\ sin(x)} \, dx[/tex]
Required
Evaluate
[tex]\int\limits {cos(x)\ sin(2x)\ sin(x)} \, dx[/tex]
Rewrite as:
[tex]\int\limits {cos(x)\ sin(2x)\ sin(x)} \, dx = \int\limits {cos(x)\ sin(x)\ sin(2x)} \, dx[/tex]
In trigonometry:
[tex]sin(2x) = 2\ sin(x)\ cos(x)[/tex]
Divide both sides by 2
[tex]\frac{1}{2}sin(2x) = \frac{2\ sin(x)\ cos(x) }{2}[/tex]
[tex]\frac{1}{2}sin(2x) = sin(x)\ cos(x)[/tex]
[tex]\frac{1}{2}sin(2x) = cos(x)\ sin(x)[/tex]
Substitute [tex]\frac{1}{2}sin(2x)[/tex] for [tex]cos(x)\ sin(x)[/tex]
[tex]\int\limits {cos(x)\ sin(2x)\ sin(x)} \, dx = \int\limits {\frac{1}{2}sin(2x)\ sin(2x)} \, dx[/tex]
[tex]\int\limits {cos(x)\ sin(2x)\ sin(x)} \, dx = \int\limits {\frac{1}{2}sin^2(2x)} \, dx[/tex]
[tex]\int\limits {cos(x)\ sin(2x)\ sin(x)} \, dx = \frac{1}{2}\int\limits {sin^2(2x)} \, dx[/tex]
Let [tex]u = 2x[/tex]
Differentiate:
[tex]du = 2 \ dx[/tex]
Make [tex]dx[/tex] the subject
[tex]dx = \frac{1}{2}du[/tex]
Substitute [tex]\frac{1}{2}du[/tex] for [tex]dx[/tex]
[tex]\int\limits {cos(x)\ sin(2x)\ sin(x)} \, dx = \frac{1}{2}\int\limits {sin^2(2x)} \, \frac{1}{2}du[/tex]
Substitute 2x for u
[tex]\int\limits {cos(x)\ sin(2x)\ sin(x)} \, dx = \frac{1}{2}\int\limits {sin^2(u)} \, \frac{1}{2}du[/tex]
[tex]\int\limits {cos(x)\ sin(2x)\ sin(x)} \, dx = \frac{1}{2}*\frac{1}{2}\int\limits {sin^2(u)} \, du[/tex]
[tex]\int\limits {cos(x)\ sin(2x)\ sin(x)} \, dx = \frac{1}{4}\int\limits {sin^2(u)} \, du[/tex]
At this point, we apply the reduction formula:
Which is:
[tex]\int\limits {sin^n(u)} \, du = \frac{n-1}{2}\int\limits sin^{n-2}(u)\ du\ - \frac{cos(u)sin^{n-1}(u)}{n}\du[/tex]
Let n = 2; So, we have:
[tex]\int\limits {sin^2(u)} \, du = \frac{2-1}{2}\int\limits sin^{2-2}(u)\ du\ - \frac{cos(u)sin^{2-1}(u)}{2}\du[/tex]
[tex]\int\limits {sin^2(u)} \, du = \frac{2-1}{2}\int\limits sin^{0}(u)\ du\ - \frac{cos(u)sin^{2-1}(u)}{2}\du[/tex]
[tex]\int\limits {sin^2(u)} \, du = \frac{1}{2}\int\limits sin^{0}(u)\ du\ - \frac{cos(u)sin^{2-1}(u)}{2}\du[/tex]
[tex]sin^0(u) = 1[/tex]
So, we have:
[tex]\int\limits {sin^2(u)} \, du = \frac{1}{2}\int\limits 1\ du\ - \frac{cos(u)sin^{2-1}(u)}{2}\du[/tex]
Integrate 1 with respect to u
[tex]\int\limits {sin^2(u)} \, du = \frac{1}{2}u - \frac{cos(u)sin^{2-1}(u)}{2}\du[/tex]
[tex]\int\limits {sin^2(u)} \, du = \frac{1}{2}u - \frac{cos(u)sin(u)}{2}\du[/tex]
Recall that:
[tex]\int\limits {cos(x)\ sin(2x)\ sin(x)} \, dx = \frac{1}{4}\int\limits {sin^2(u)} \, du[/tex]
So, we have:
[tex]\int\limits {cos(x)\ sin(2x)\ sin(x)} \, dx = \frac{1}{4}[ \frac{1}{2}u - \frac{cos(u)sin(u)}{2}\du][/tex]
Open bracket
[tex]\int\limits {cos(x)\ sin(2x)\ sin(x)} \, dx = \frac{1}{8}u - \frac{cos(u)sin(u)}{8}[/tex]
Recall that: [tex]u = 2x[/tex] and [tex]du = 2 \ dx[/tex] [tex]dx = \frac{1}{2}du[/tex]
So, the expression becomes:
[tex]\int\limits {cos(x)\ sin(2x)\ sin(x)} \, dx = \frac{1}{8}2x - \frac{cos(2x)sin(2x)}{8}[/tex]
[tex]\int\limits {cos(x)\ sin(2x)\ sin(x)} \, dx = \frac{1}{4}x - \frac{cos(2x)sin(2x)}{8}[/tex]
Add constant c
[tex]\int\limits {cos(x)\ sin(2x)\ sin(x)} \, dx = \frac{1}{4}x - \frac{cos(2x)sin(2x)}{8} +c[/tex]
----------------------------------------------------------------------------------------
In trigonometry:
[tex]sin(2\theta) = 2sin(\theta)cos(\theta)[/tex]
Divide both sides by 2
[tex]\frac{1}{2}sin(2\theta) = \frac{2sin(\theta)cos(\theta)}{2}[/tex]
[tex]\frac{1}{2}sin(2\theta) = sin(\theta)cos(\theta)[/tex]
Replace 2x with [tex]\theta[/tex]
[tex]\frac{1}{2}sin(2*2x) = sin(2x)cos(2x)[/tex]
[tex]\frac{1}{2}sin(4x) = sin(2x)cos(2x)[/tex]
----------------------------------------------------------------------------------------
[tex]\int\limits {cos(x)\ sin(2x)\ sin(x)} \, dx = \frac{1}{4}x - \frac{cos(2x)sin(2x)}{8} +c[/tex] becomes
[tex]\int\limits {cos(x)\ sin(2x)\ sin(x)} \, dx = \frac{1}{4}x - \frac{sin(4x)}{2*8} +c[/tex]
[tex]\int\limits {cos(x)\ sin(2x)\ sin(x)} \, dx = \frac{1}{4}x - \frac{sin(4x)}{16} +c[/tex]
[tex]\int\limits {cos(x)\ sin(2x)\ sin(x)} \, dx = \frac{x}{4} - \frac{sin(4x)}{16} +c[/tex]
The solution can be further simplified as:
[tex]\int\limits {cos(x)\ sin(2x)\ sin(x)} \, dx = \frac{4x-sin(4x)}{16} +c[/tex]
X^2- 2x - 54 = 0
I need the answer for this please
x = -6, 8
Find the values of x by factoring, getting
(x - 8)(x + 6) = 0
Answer:
x = -6, 8
Step-by-step explanation:
The value of a painting has a growth factor of 107. The current value of the painting is $345. How much was it orginally worth when you purchased it 10 years ago
Answer:
70$
Step-by-step explanation:
Could someone help me that knows and won't put a bad answer. 30 points.
Answer:
line D
Step-by-step explanation:
The slope of the line D is y= 1/2x-1.5
what is m/a
side a
side b
A rectangle with an area of 47m² is dilated by a factor of 7. What is the area of the dilated rectangle? Enter your answer in the box. Do not leave your answer as a fraction.
?m²
Answer:
28m^2
Step-by-step explanation:
Answer:
28 square meters is correct for all future users.
Step-by-step explanation:
Determinar el complemento del ángulo 67 grados centígrados
Answer:
spooky spooky spooky I no speak Spanish
Step-by-step explanation:
same
Alexis was making cupcakes for the town bake sale. She decorated 9
cupcakes in 30 minutes. If Alexis continued decorating cupcakes at
this rate, how many cupcakes would she decorate in 2 hours?
Johnny said that Alexis would be able to decorate 18 cupcakes in 2
hours. Is Johnny correct? If not, explain his mistake and identify the
correct solution.
Answer:
Incorrect
Step-by-step explanation:
because it takes 30 min to bake 9 which is half an hour, and if if you multiply 9 by 4 which 30 mins each it would be 36 cupcakes in 2 hours.
On Saturday, Mark sold 2 7/8 gallons of lemonade.
On the same day, Regan sold 2/3 as much lemonade as Mark.
How much lemonade, in gallons, did Regan sell?
Answer:
Regan sold [tex]\mathbf{\frac{23}{12}}[/tex] gallons of lemonade
Step-by-step explanation:
Operations with Fractions
Mark sold [tex]2\frac{7}{8}[/tex] gallons of lemonade.
Regan sold [tex]\frac{2}{3}[/tex] as much lemonade as Mark
We can find the portion of lemonade Regan sold by multiplying both fractions. But first, we must convert to improper fraction:
[tex]2\frac{7}{8}=2+\frac{7}{8}=\frac{23}{8}[/tex]
Now we multiply:
[tex]\frac{23}{8}\cdot \frac{2}{3}=\frac{46}{24}=\frac{23}{12}[/tex]
Regan sold [tex]\mathbf{\frac{23}{12}}[/tex] gallons of lemonade
010. Tyler's brother works in a shoe store. Tyler's brother earns a commission. He
makes 2.5% of the amount he sells. Last week, he sold $800 worth of shoes. How
much was his commission?
PLSS I NEED HELP
Answer:
$20
Step-by-step explanation:
800 × 2.5% = $20
Glad I could help!
HELP FAST PLSSSS IM DESPERATE RLLY
I will give someone 100 points if they give me a brainliest
Answer:
ok
but why?
hmm
I REALLY NEED HELP ITS WORTH 30 POINTS
Answer:
1) x < 30
2) x > 20
3) x >= 250
4) x > 20
5-10) I cant see the numbers on the graph
Step-by-step explanation:
Answer:
For the graphs you have to basically make those words into a mathematical equation to make more sense like for the first problem in which..
x>30
remember, open circles mean that its it can't be greater for symbols like < and >
closed circles are for inequalities that are greater than or equal to or less than or equal to which are these symbols: ≤ and ≥
what is 250,000 divided by 12
Answer:
20833.3
Step-by-step explanation:
Answer:
tthe answer is. 20833.3
The population of Buhlerville is 92,000 in 2019. The population of the city was 94880 in 2021. What is the
expected population in 2022?
Answer:
96320
Step-by-step explanation:
The population of Buhlerville will be 96320
help!!! Pleasee thank youuuu
Answer:
a
Step-by-step explanation:
it is the reflection or mirror image
What is the value of r of the geometric series?
Sigma-Summation Underscript n = 1 Overscript 3 EndScripts 1.3 (0.8) Superscript n minus 1
Answer:
A) 0.8
Step-by-step explanation:
0.8 on edge.
The value of r of the geometric series is 4/5.
What is geometric series?
The geometric series defined as a series represents the sum of the terms in a finite or infinite geometric sequence. The successive terms in this series share a common ratio.
The nth term of a geometric progression is expressed as
Tₙ = arⁿ⁻¹
Where a is the first term, r is the common ratio
Given the formula for the geometric series expressed as
[tex]_3\\\sum{1.3(0.8)^n^-^1}\\n=1[/tex]
Comparing both equations
rⁿ⁻¹ = (0.8)ⁿ⁻¹
r = 0.8
r = 8/10
r = 4/5
Hence, the common ratio is 4/5.
Learn more about geometric series here:
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PLS HELP I ONLY HAVE UNTIL TOMORROW, I WILL GIVE BRAINLIEST TO WHO ANSWERS CORRECTLY AND EXPLAINS
How do i draw a growth triangle for the problem y= 2x-3? I have to find M and B
Arthur walked 6 kilometers around the park, and D.W. sprinted 200 m at the park. How many meters did they travel in total?
Answer:
6200 meters
Step-by-step explanation:
Answer:
Step-by-step explanation:
The answer should be 6200
Adrienne has chickens and ducks on her farm. She has 48 chickens and 28 ducks. What ratio describes the number of ducks to the number of chickens?
At a pet store, a survey was taken asking how many pets each person had. The
results were: (2,5, 3, 1, 0, 4, 2, 7,0, 2, 7,3). What is the mode?
Answer:
2
Step-by-step explanation:
We have these numbers: 2, 5, 3, 1, 0, 4, 2, 7, 0, 2, 7, 3
The mode is the most common value. I can see 3 two's in the data set. The mode is 2.
#SpreadTheLove <3
f(x) = x^2 -2x -3 /x-3 identify the discontinuity
Answer:
Step-by-step explanation:
Again, parentheses would help with clarity. f(x) = (x^2-2x-3)/(x-3)
The denominator is x-3. Division by 0 is undefined, so x ≠ 3.
If the Mustangs, the Braves, the Bulldogs, the Huskies, and the Vikings are playing in the same sports league,
in how many different orders can they finish their season?
Answer:
.
Step-by-step explanation:
Slope please help me find it need done today
Answer:
5. - 1/9
6. 4
7. 5
8. 5
Step-by-step explanation:
What is the area of the trapezoid?
Answer:
A=78ft
Step-by-step explanation:
The area of a trapezoid is A=(a+b)/2 *h
a=long base
b=short base
h=height
For your trapezoid, it will look like this:
a=15ft
b=11ft
h=6ft
Let's add those into the equation:
A=(15+11)/2 *6
Okay, let's solve it! Using PEMDAS, we know parenthesis are first.
A=26/2 *6
Now we divide!
A=13*6
Multiply! And don't forget to add your unit.
A=78ft
A fitness club charges a onetime $75 initiation fee plus $30 each month for a member to maintain her membership. Which equation below models the total cost, c, the club charges as a function of the number of months, m, that the individual has maintained her membership?
1) c = 75m + 30
2) c = 1/30m + 75
3) c = 30m + 75
4) c = 900 + 30m
Answer:
c = 30m + 75
Step-by-step explanation:
Because the 75 dollars is only one time you add that to the total. Because it is 30 dollars PER MONTH and m is the number of months you would multiply those two in order to know how much you would be paying!
Hope this helps!
What is equivalent to 3;4
Answer:
6;8
3;4 that the lowest that it can go. the equivalent is 6;8 but if you want to change it into decimal form you can divide by two which is 1.5;2 but the answer is 6;8 because 3;4 is already in it's lowest form, or is already simplified to it's maximum extent.
Look at the triangles shown below.
z
2
6
E
А
3 D
Which statement is true?
(Ity)
O SinceAAEC ~AADB according to the side-angle-side similarity postulate, then
2 6
(aty)
SinceAAEC ~AADB according to the side-angle-side similarity postulate, then
2
9 3
o SinceAAEC ~AADB according to the angle-angle similarity postulate, then
(1+y)
6 3
o Since AAEC ~AADB according to the angle-angle similarity postulate, then
(1+y)
NOH OH
T
Answer:c
Step-by-step explanation:
The statement which is true for the given expression triangle is, 9/(x + y) = 3/x. So option c is correct.
What is similarity of triangles?Triangles with the same shape but different sizes are said to be similar triangles. Squares with any side length and all equilateral triangles are examples of related objects. In other words, if two triangles are similar, their corresponding sides are proportionately equal and their corresponding angles are congruent. Triangle resemblance is shown here by the symbol "≈"
Given that,
One right angle triangle, ΔAEC
one segment BD which is intersecting sides AC & AE
ΔAEC has one similar triangle inside, which is ΔADB
Triangle ΔAEC and ΔADB similar,
So, we can say ΔAEC ≈ ΔADB
For the given similar triangles,
we can use angle-angle similarity property,
so, AE/AC = AD/AB
⇒ (3+6)/(x + y) = 3/x
⇒9/(x + y) = 3/x
Hence, the Statement C is correct
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3.
The students in a class were surveyed to find out how many people live in their households.
The results of the survey are shown in the dot plot below.
Number of People in Household
What percent of the class has a household of 7 or 8 people?
Answer:
it is number 4
Step-by-step explanation:
0.75 times what equals 3
What is the answer
Answer:
4
3/.75=4
divide 3 by point seven five
