Mathematics College

Evaluate the integral. (Remember to use absolute values where appropriate. Use C for the constant of integration.) cos(x) sin(2x) sin(x) dx

Answers

Answer 1

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]

Related Questions

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

Answers

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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PLS HELP I ONLY HAVE UNTIL TOMORROW, I WILL GIVE BRAINLIEST TO WHO ANSWERS CORRECTLY AND EXPLAINS

Answers

i’ve seen this exact question so many times from other people

What is equivalent to 3;4

Answers

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.

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

Answers

Answer:

28m^2

Step-by-step explanation:

Answer:

28 square meters is correct for all future users.

Step-by-step explanation:

X^2- 2x - 54 = 0



I need the answer for this please

Answers

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 population of Buhlerville is 92,000 in 2019. The population of the city was 94880 in 2021. What is the
expected population in 2022?

Answers

Answer:

96320

Step-by-step explanation:

The population of Buhlerville will be 96320

f(x) = x^2 -2x -3 /x-3 identify the discontinuity

Answers

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.

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.

Answers

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.

about 36 cupcakes in two hours, johnny is incorrect because it would be 18 cupcakes per ONE hour.

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?

Answers

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

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

Answers

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!

HELP FAST PLSSSS IM DESPERATE RLLY

Answers

The answer is y= 1/3 x -1
Answer : y =1/3 x -1

I REALLY NEED HELP ITS WORTH 30 POINTS

Answers

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 ≥

How do i draw a growth triangle for the problem y= 2x-3? I have to find M and B

Answers

Well m is 2 and b is 3
Using the equation y=Mx+B

I will give someone 100 points if they give me a brainliest

Answers

Answer:

ok

but why?

hmm

i dont get it , you will give someone 100 points for brainlist ? just answer someone’s question and ask for brainliest

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?

Answers

Answer:

.

Step-by-step explanation:

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?

Answers

Answer:

it is number 4

Step-by-step explanation:

help!!! Pleasee thank youuuu

Answers

Answer:

a

Step-by-step explanation:

it is the reflection or mirror image

Slope please help me find it need done today

Answers

Answer:

5. - 1/9

6. 4

7. 5

8. 5

Step-by-step explanation:

Could someone help me that knows and won't put a bad answer. 30 points.

Answers

Answer:

line D

Step-by-step explanation:

The slope of the line D is y= 1/2x-1.5

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? ​

Answers

12 chickens to 7 ducks 12:7

what is 250,000 divided by 12

Answers

Answer:

20833.3

Step-by-step explanation:

Answer:

tthe answer is. 20833.3

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

Answers

Answer:

$20

Step-by-step explanation:

800 × 2.5% = $20

Glad I could help!

Determinar el complemento del ángulo 67 grados centígrados

Answers

Answer:

spooky spooky spooky I no speak Spanish

Step-by-step explanation:

same

what is m/a
side a
side b

Answers

side b side b side b side b

What is the area of the trapezoid?

Answers

78. you can find the area of the two ends which are triangles by subtracting 11 from 15. Then divide that four by two since both ends are included. Multiply 2 by 6 to get half the area of one triangle. But since there are two just leave that number as is. Finally multiply 11 by 6 to get the rest of the area.

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

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

Answers

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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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

Answers

Answer:

70$

Step-by-step explanation:

0.75 times what equals 3

What is the answer

Answers

4 bc .75 x4 is 3 the answer is 4

Answer:

4

3/.75=4

divide 3 by point seven five

Arthur walked 6 kilometers around the park, and D.W. sprinted 200 m at the park. How many meters did they travel in total?

Answers

Answer:

6200 meters

Step-by-step explanation:

Answer:

Step-by-step explanation:

The answer should be 6200

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?

Answers

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

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