Answer:
[tex]\displaystyle \int \sec\left(\frac{x}{2}\right)\tan^5\left(\frac{x}{2}\right)dx=\frac{2\sec^5\left(\dfrac{x}{2}\right)}{5}-\frac{4\sec^3\left(\dfrac{x}{2}\right)}{3}+2\sec\left(\frac{x}{2}\right)+C[/tex]
Step-by-step explanation:
We want to find the integral:
[tex]\displaystyle \int \sec\left(\frac{x}{2}\right)\tan^5\left({\frac{x}{2}\right)dx[/tex]
First, perform the substitution y = x / 2. This yields:
[tex]\displaystyle dy = \frac{1}{2} \, dx \Rightarrow dx = 2\, dy[/tex]
Hence, the integral becomes;
[tex]\displaystyle =2\int \sec y\tan^5 y \, dy[/tex]
Next, as you had done, let's expand the tangent term but to the fourth:
[tex]\displaystyle =2\int \sec y\tan^4 y\tan y\, dy[/tex]
Recall that:
[tex]\tan^2(y)=\sec^2(y)-1[/tex]
Hence:
[tex]\displaystyle =2\int \sec y(\sec^2 y-1)^2\tan y\, dy[/tex]
We can use substitution once more. This time, let u = sec(y). Hence:
[tex]\displaystyle du = \sec y \tan y \, dy[/tex]
Rewrite:
[tex]\displaystyle =2\int \left((\sec^2y-1)^2\right)\left(\sec y \tan y\right)dy[/tex]
Therefore:
[tex]\displaystyle = 2\int (u^2 - 1)^2\, du[/tex]
Expand:
[tex]\displaystyle =2\int u^4-2u^2+1\, du[/tex]
Reverse Power Rule:
[tex]\displaystyle = 2\left(\frac{u^5}{5} - \frac{2u^3}{3} + u\right) + C[/tex]
Simplify:
[tex]\displaystyle = \frac{2u^5}{5} - \frac{4u^3}{3} + 2u + C[/tex]
Back-substitute:
[tex]\displaystyle = \frac{2\sec^5 y }{5}-\frac{4\sec^3 y}{3}+2\sec y+C[/tex]
Back-substitute:
[tex]\displaystyle =\frac{2\sec^5\left(\dfrac{x}{2}\right)}{5}-\frac{4\sec^3\left(\dfrac{x}{2}\right)}{3}+2\sec \left(\frac{x}{2}\right)+C[/tex]
Therefore:
[tex]\displaystyle \int \sec\left(\frac{x}{2}\right)\tan^5\left(\frac{x}{2}\right)dx=\frac{2\sec^5\left(\dfrac{x}{2}\right)}{5}-\frac{4\sec^3\left(\dfrac{x}{2}\right)}{3}+2\sec\left(\frac{x}{2}\right)+C[/tex]
Answer:
= 2 (sec (x/2) - 2sec³(x/2) + sec⁵(x/2) ) + C
3 5
Step-by-step explanation:
∫sec(x/2) tan⁵(x/2) dx
apply u substitute u = x/2
= ∫sec(u) tan⁵(u) * 2du
= 2 * ∫sec(u) tan⁴(u) tan(u) du
= 2 * ∫sec(u) (tan²(u))² tan(u) du
= 2 * ∫sec(u) (-1 + sec²(u))² tan(u) du
apply u substitute v = sec(u)
= 2 * ∫(-1 + v²)² dv
expand
= 2 * ∫1 - 2v² + v⁴ dv
sum
= 2 (v - 2v³ + v⁵ )
3 5
substitute it back
= 2 (sec (x/2) - 2sec³(x/2) + sec⁵(x/2) )
3 5
add constant to the solution.
= 2 (sec (x/2) - 2sec³(x/2) + sec⁵(x/2) ) + C
3 5
I'm trying to find sin of (7pi/12) exactly using an angle of addition or subtraction formula could anyone help me
Answer:
(√2 + √6)/4.
Step-by-step explanation:
Use the addition formula
sin (A+ B) = sinAcosB + cosAsinB
sin (7pi/12) = sin( 3pi/12 + 4pi/12)
= sin(pi/4 + pi/3) = sin pi/4 cos pi/3 + cos pi/4 sin pi/3
= 1/√2 * 1/2 + 1/√2 * √3/2 (using the 45-45-90 and 30-60-90 triangles)
= 1 / 2√2 + √3/2√2
= √2/4 + √6/4
= (√2 + √6)/4
Can someone plz tell me if I’m right?
Answer:
Step-by-step explanation:
Since x is used a lot as a variable in algebra, I would use another symbol for multiplication. Though if your teacher requires you to use x as a multiplication symbol, then keep it as is. I use the asterisk symbol
Example: 2 times 3 = 2*3
------------------------------------------
For problem 2, you have the parenthesis in the wrong spots. Saying "the sum of four plus five" means we have 4+5 as you'd expect. Then you multiply that group with 2. So you'd really have (4+5)*2
How is this different from 4+5*2 back in problem 1? It comes down to how the order of operations handles things. To evaluate 4+5*2, we multiply first, then add. So we have 4+5*2 = 4+10 = 14
With the other expression, we add first because it is in the parenthesis block. Afterward we multiply the values to get (4+5)*2 = 9*2 = 18
Find The Area Of The Shape Shown Below
Answer:42.55
Step-by-step explanation:
Les get the big boi out of the way first. We see that it is 3.5 by 9 and if we multiply we get 31.5. Next left triangle 2 by 2 so four but divided by 2 is 2. God so many 2's. So total is 33.5 so far. Next triangle is 2 by 5 soo 10 divided by 2 is 5. total is 38.5. Last dude in the middle. We know one side is two so we have to subtract here from the triangles which gets u the other side of 2 so 4. Total is 42.5
The Area of the Shape Shown is 42.5 square units.
What is Area of Rectangle?The area of Rectangle is length times of width.
The area of the rectangle in the given figure is
Area of rectangle=3.5×9
=31.5 square units.
The area of left side triangle is
Area of triangle=1/2×2×2
=2 square units
The area of right side triangle 1/2×5×2
=5 square units
Now the area of square =2²
=4 square units.
Now total area of figure is 31.5+2+5+4 is 42.5 square units.
Hence, the Area of the shape Shown is 42.5 square units.
To learn more on Area of Rectangle click:
https://brainly.com/question/20693059
#SPJ2
A bag contains 6 RED beads, 3 BLUE beads, and 11 GREEN beads. If a single bead is picked at random, what is the probability that the bead is RED or GREEN?
Answer:
17/20
Step-by-step explanation:
Total no. of beads = 6 + 3 + 11 = 20
We need RED or GREEN bead .
So no. of beads needed = 6 + 11 = 17
So probability of getting GREEN or RED beads = 17/20
What is (x+5)(x-4)=0 Expand and solve for x
step 1
(x - 4) • (x + 5) = 0
STEP
:
A product of several terms equals zero.
When a product of two or more terms equals zero, then at least one of the terms must be zero.
We shall now solve each term = 0 separately
Solving a Single Variable Equation:
2.2 Solve : x-4 = 0
Add 4 to both sides of the equation :
x = 4
Solving a Single Variable Equation:
2.3 Solve : x+5 = 0
Subtract 5 from both sides of the equation :
x = -5
.
Answer:
A) [tex]\huge\boxed{\sf x^2 + x - 20 = 0}[/tex]
B) [tex]\huge\boxed{\sf x = - 5 \ \ \ \ OR \ \ \ \ x = 4}[/tex]
Step-by-step explanation:
[tex]\sf (x+5)(x-4) = 0[/tex]
[tex]\rule[225]{225}{2}[/tex]
Expanding:
Expanding Brackets
[tex]\sf x^2 -4x+5x-20 = 0[/tex]
Adding / Subtracting Like terms
[tex]\sf x^2 + x - 20 = 0[/tex]
[tex]\rule[225]{225}{2}[/tex]
Solving:
Given the equation:
(x+5)(x-4) = 0
Using zero Method
Either,
x + 5 = 0 OR x - 4 = 0
x = - 5 OR x = 4
[tex]\rule[225]{225}{2}[/tex]
Given the following formula, solve for I.
P =
2(1 + b)
Answer: [tex]l=\dfrac{P}{2}-b[/tex]
Step-by-step explanation:
P = 2(l + b)
Divide both sides by 2:
[tex]\dfrac{P}{2}=l+b[/tex]
Subtract b from both sides:
[tex]\dfrac{P}{2}-b=l[/tex]
Note: You can also multiply b by [tex]\frac{2}{2}[/tex] if you want the left side to be one fraction.
[tex]\dfrac{P}{2}-\bigg(\dfrac{2}{2}\bigg)b=l\\\\\\\dfrac{P-2b}{2}=l[/tex]
find (f+g) (x) f (x) = 4x - 4 and g (x) = 2x^2 - 3x
(f+g)(x) = f(x) + g(x)
= 4x - 4 + 2[tex]x^{2}[/tex] - 3x
= 2[tex]x^{2}[/tex] + x - 4
02.01)Which is the ratio of the number of months that begin with the letter J to the total number of months in a year? 12 to 3 3 to 9 9 to 12 3 to 12 HELP!!! FOR MY MATH TEST
Answer:
3 to 12
Step-by-step explanation:
3 months (January, June, July) and 12 monthes
Answer: the answer is 3 to 12
Step-by-step explanation: hope this will help
alex buys six equally priced candy bars for $9:00 what is the unit rate?
Answer:
$1.5 unit
Step-by-step explanation:
According to the given situation, the calculation of the unit rate is shown below:-
Candy bars = 6
Amount of candy bars = $9:00
Unit rate
[tex]= \frac{Amount\ of\ candy\ bars}{Number\ of\ candy\ bars}[/tex]
Now we will put the values into the above formula
[tex]= \frac{\$9:00}{6}[/tex]
Which gives result
= $1.5 unit
Therefore for computing the unit rate we simply divide the amount of candy bars by the number of candy bars.
(!20 POINTS PLEASE HELP!) The students in a class are randomly drawing cards numbered 1 through 28 from a hat to determine the order in which they will give their presentations. Find each probability. Solve each problem below 1. P(13) 2. P(less than 14) 3. P(not 2 or 17)
Since there is 28 cards, the chance of drawing 1 card will be [tex]\frac{1}{28}[/tex]
Therefore, [tex]P(13)=\frac{1}{28}[/tex].
There are 13 numbers less than 14 and greater than 0.
Therefore [tex]P(>14)=\frac{13}{28}[/tex]
There are 26 numbers that are not 2 or 17.
Therefore [tex]P(\neq 2,17)=\frac{26}{28}=\frac{13}{14}[/tex]
Hope this helps.
頑張って!
Probability of
Option (1). P(13) =[tex]\frac{1}{28}[/tex]
Option (2). P(less than 14) = [tex]\frac{13}{28}[/tex]
Option (3). P(not 2 or 17) =[tex]\frac{13}{14}[/tex]
What is Probability?Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur
Given,
Total number of card = 28
Probability = Number of favorable outcome / Number of total outcome
Probability of picking 13 = [tex]\frac{1}{28}[/tex]
Probability of picking a card less then 14 = [tex]\frac{13}{28}[/tex]
Probability of not picking 2 or 17 = [tex]\frac{26}{28}=\frac{13}{14}[/tex]
Hence, the probability of
Option (1). P(13) =[tex]\frac{1}{28}[/tex]
Option (2). P(less than 14) = [tex]\frac{13}{28}[/tex]
Option (3). P(not 2 or 17) =[tex]\frac{13}{14}[/tex]
Learn more about Probability here
https://brainly.com/question/11234923
#SPJ2
please help me i would appreciate it
Answer:
1/2 meter long
Step-by-step explanation:
To answer this question use the equation 1 1/3 - 5/6
Next make 1 1/3 an improper fraction
To do this multiply the integer and the denominator and then add all of that plus the numerator
This means that 1 1/3 is 4/3
4/3 - 5/6
Next make the denominators the same
This means finding the LCM which is 6
8/6 - 5/6 = 3/6 (1/2)
Heidis hair is 1/2 meters long
A couple wants to install a square mirror on their bathroom wall. The area of the square mirror is 720 square inches. To the nearest hundredth of an inch, what length of wood trim is needed to go around the entire mirror?
Answer:
107.33 inches
Step-by-step explanation:
Area of the square more = length ^2
Area of the square mirror = 720 square inches
Area of the square more = length ^2
720 = length^2
Find the square root of both sides
√720 = √lenght^2
Length = √720
Length = 26.83281573 inches
Each length of the wood = 26.83281573 inches
There are 4 equal sides on the square mirror
Total length of the wood = 26.83281573 × 4
= 107.33126292 inches
Length of wood trim is needed to go around the entire mirror to the nearest hundredth = 107.33 inches
T is the midpoint of SU, ST = 8x + 11 and TU = 12x-1, find the value of x.
Answer:
x = 3
Step-by-step explanation:
mid point means it bisects the segment into two equals halfs meaning
ST = TU
8x + 11 = 12x - 1
8x + 12 = 12x
12 = 4x
3 = x
Amy, Ray, Kim, and Jamal are on a trivia team. They gain points for correct answers
and lose points for incorrect answers. During the contest, Amy gains 3 points, Ray loses
4 points, Kim loses 2 points, and Jamal gains 5 points. How many points does the team
have at the end of the contest?
Answer:
2 points
Step-by-step explanation:
0+3-4-2+5=2 points
do all sets have subsets?
Answer:
yes
Step-by-step explanation:
d) 256 chocolates are distributed among the students of class 5.
If the number of chocolates obtained by each students is the same
as the number of students in the class, find the number of students
in the class?
Answer:
16
Step-by-step explanation:
Let x be the number of chocolates obtained by each student
Since the number of chocolates obtained by each student = the number of students in the class, then, number of students in the class = x
256/x = x
Multiply both sides by x
256 = x × x ( x squared)
Multipy powers raised on both sides by 1/2
X = 16
y varies inversely with x. If y = 5 when x = 60, find y when x = 150
pre cal
Answer:
y = 2 when x = 150
Step-by-step explanation:
Inverse variation is
xy = k
5*60 = k
300 = k
xy = 300
150y = 300
Divide by 150
150y/150 = 300/150
y = 2
Integrate the following w.r.t x 1) 2x^2/3.
2) (5-x)^23
Answer:
A) [tex]\int\frac{2x^2}{3}dx=\frac{2x^3}{9}+C[/tex]
B) [tex]\int(5-x)^{23}dx=-\frac{(5-x)^{24}}{24}+C[/tex]
Step-by-step explanation:
A)
So we have the integral:
[tex]\int\frac{2x^2}{3}dx[/tex]
First, remove the constant multiple:
[tex]=\frac{2}{3}\int x^2\dx[/tex]
Use the power rule, where:
[tex]\int x^ndx=\frac{x^{n+1}}{x+1}[/tex]
Therefore:
[tex]\frac{2}{3}\int x^2\dx\\=\frac{2}{3}(\frac{x^{2+1}}{2+1})[/tex]
Simplify:
[tex]=\frac{2}{3}(\frac{x^{3}}{3})[/tex]
And multiply:
[tex]=\frac{2x^3}{9}[/tex]
And, finally, plus C:
[tex]=\frac{2x^3}{9}+C[/tex]
B)
We have the integral:
[tex]\int(5-x)^{23}dx[/tex]
To solve, we can use u-substitute.
Let u equal 5-x. Then:
[tex]u=5-x\\du=-1dx[/tex]
So:
[tex]\int(5-x)^{23}dx\\=\int-u^{23}du[/tex]
Move the negative outside:
[tex]=-\int u^{23}du[/tex]
Power rule:
[tex]=-(\frac{u^{23+1}}{23+1})[/tex]
Add:
[tex]=-(\frac{u^{24}}{24})[/tex]
Substitute back 5-x:
[tex]=-(\frac{(5-x)^{24}}{24})[/tex]
Constant of integration:
[tex]=-\frac{(5-x)^{24}}{24}+C[/tex]
And we're done!
Bolts are packed into bags at a factory. Each bag should have 25 bolts in it but a bad with 22 or 28 is acceptable. Formulate an absolute value equation that could be used to solve for the minimum and maximum number of bolts in a bag.
|x____ | = ______
x = _____(smaller number here)
x = ________(larger number here)
Blank 1:
Blank 2:
Blank 3:
Blank 4:
Answer:
Absolute value equation is; |x - 25| ≤ 3
Minimum number of bolts is 22 while Maximum number of bolts is 28
Step-by-step explanation:
We are told that;
Each bag should have 25 bolts in it but 22 or 28 is acceptable.
Since 22 or 28 is acceptable, it means the allowable error is ±3.
Thus, if x is the number of bolts in the bag, the absolute value equation would be;
|x - 25| ≤ 3
Minimum number of bolts is 22 while maximum is 28
Angles 1 and 2 are complementary, and angle 1=3x+3 and angle 2=10x-4, find the degree measure of each angle
Answer:
The measure of the angles:
24° and 66°
Step-by-step explanation:
Complementary angles sum 90°
then:
(3x + 3) + (10x-4) = 90
3x + 10x + 3 - 4 = 90
13x - 1 = 90
13x = 90+1
13x = 91
x = 91/13
x = 7°
then:
angle 1:3x+3 = 3*7 + 3 = 21+3 = 24°
angle 210x - 4 = 10*7 - 4 = 70 - 4 = 66°
Check:
66° + 24° = 90°
use the values in the table to determine the slope.
Answer:
-3/2
Step-by-step explanation:
Take two points from the table and use the slope formula
m= (y2-y1)/(x2-x1)
= ( 19 - 13)/ ( -4 - 0)
= 6/-4
= -3/2
CAN SOMONE PLEASE HELP ME ASAP PLEASEEE!!!
Answer:
The yellow polygon is the scaled version of the red one (scaled by 3×), so the variable w = 9
Answer:
9
Step-by-step explanation:
The similarity ratio is 4/x = z/9 = 2/6 = 3/w = y/15
2/6 = 3/w cross multiply expressions 2w = 18 and w = 9
A device has a constant failure rate with a MTTF of 2 months. One hundred of the devices are tested to failure. (a) How many of the devices do you expect to fail during the second month
Answer: the number of devices expected to fail during the second month is 24
Step-by-step explanation:
Given that
The device has a constant failure rate with MTTF of 2 months.
As the device has constant failure rate so it has exponential failure distribution
f(t) = λe^-λt
Here MTTF = 1/ λ
so λ = 1/2 Months⁻¹ = 0.5 Months⁻¹ and from the question, Number of devices = 100
E( 1 < x < 2) = E ( x < 2) - E (x < 1)
so E(x < X) can be calculated with λ = 0.5 Months⁻¹ will be calculated as the failure function
f(x) = λ exp ( - λ×t) for t > 0
F (x>0) = 1 - exp( - λx)
so E ( 1 < x < 2) = E ( x < 2) - E (x < 1)
E ( x < 2) = 1 - exp(-0.5 × 2) = 0.6321 ; E (x<1) = 1 - exp(-0.5 × 2) = 1 - exp( -0.5) = 0.3934
so E ( 1 < x < 2) = 0.6321 - 0.3934 = 0.2387
so the number of devices expected to fail during the second month is;
100 × 0.2387 = 23.87 ≈ 24
Find FH
Need help ASAP !!
Answer:
[tex]FH=22[/tex]
Step-by-step explanation:
FH is the combined lengths of FG and GH.
In an equation, this is:
[tex]FH=FG+GH[/tex]
We already know that FG is 8 and that GH is 14. Thus:
[tex]FH=8+14[/tex]
Add:
[tex]FH=22[/tex]
So, the length of FH is 22.
And we're done!
Write the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 in this order and insert '+' or '-' between them to get the result 3? Please Help
Answer:
0 - 1 - 2 + 3 - 4 + 5 - 6 +7 - 8 + 9 = 3
Step-by-step explanation:
Glados
spent the day at the mall, First, she bought
three bikes for $10 each. Later, she found 22
dollar bills. Write the total change
an integer.
to Glado's
funds as an integer
Answer:
8
Step-by-step explanation:
When you woke up this morning, the temperature was -5.8°C. At noon, the temperature was 2.9°C. Which expression and statement describes the situation?
Answer:
-5.8<2.9
Step-by-step explanation:
Malik borrowed $105 from his brother and plans to repay $12 per week. He creates a table to help him determine the
number of weeks it will take to repay his brother. How many weeks will it take Malik to repay his brother?
8
9
10
11
please help!!!!
Answer:
9
Step-by-step explanation:
12×9=108 108>105 so it can't be any other number cuz we go too high or too low
Answer:
9
Step-by-step explanation:
Find an equation of the plane.The plane through the point(4, −3, −1)and parallel to the plane 6x − y − z = 6
Both planes have the same normal vector:
[tex]6x-y-z=6\implies\mathbf n=\langle6,-1,-1\rangle[/tex]
The plane we want must contain the point (4, -3, -1), so its equation would be
[tex]\mathbf n\cdot\langle x-4,y+3,z+1\rangle=0[/tex]
[tex]\implies 6(x-4)-(y+3)-(z+1)=0[/tex]
[tex]\implies\boxed{6x-y-z=28}[/tex]
A factory worker can produce 28 toys in an eight hour day If there are 42 workers in total how many toys are produced every hour
Answer:
147 toys
Step-by-step explanation:
for 1 person, 28toys/8hours = 3.5toys/ 1hour
for 42 people, 42 * 3.5toys/ 1hour = 147 toys/hour
~Hope it Helps!~