Replace x with π/2 - x to get the equivalent integral
[tex]\displaystyle \int_{-\frac\pi2}^{\frac\pi2} \cos(\cot(x) - \tan(x)) \, dx[/tex]
but the integrand is even, so this is really just
[tex]\displaystyle 2 \int_0^{\frac\pi2} \cos(\cot(x) - \tan(x)) \, dx[/tex]
Substitute x = 1/2 arccot(u/2), which transforms the integral to
[tex]\displaystyle 2 \int_{-\infty}^\infty \frac{\cos(u)}{u^2+4} \, du[/tex]
There are lots of ways to compute this. What I did was to consider the complex contour integral
[tex]\displaystyle \int_\gamma \frac{e^{iz}}{z^2+4} \, dz[/tex]
where γ is a semicircle in the complex plane with its diameter joining (-R, 0) and (R, 0) on the real axis. A bound for the integral over the arc of the circle is estimated to be
[tex]\displaystyle \left|\int_{z=Re^{i0}}^{z=Re^{i\pi}} f(z) \, dz\right| \le \frac{\pi R}{|R^2-4|}[/tex]
which vanishes as R goes to ∞. Then by the residue theorem, we have in the limit
[tex]\displaystyle \int_{-\infty}^\infty \frac{\cos(x)}{x^2+4} \, dx = 2\pi i {} \mathrm{Res}\left(\frac{e^{iz}}{z^2+4},z=2i\right) = \frac\pi{2e^2}[/tex]
and it follows that
[tex]\displaystyle \int_0^\pi \cos(\cot(x)-\tan(x)) \, dx = \boxed{\frac\pi{e^2}}[/tex]
A closet contains 5 cases of laundry detergent. Each case contains 12 bottles. Each bottle contains 3 liters of detergent.
How many liters of laundry detergent in all are in the closet?
180 L
63 L
41 L
20 L
Answer:
A
Step-by-step explanation:
Find the number of bottles in 5 cases:
5 × 12 = 60
Find the number of liters in 60 bottles:
60 × 3 = 180
180 liters of laundry detergent
The polynomial below is a perfect square trinomial of the form A^2 - 2AB + B^2
9x^2 - 18x + 36
A. True
B. False
Answer:
False
Step-by-step explanation:
[tex]9x^2-18x+36\\\\=(3x)^2- 3x \cdot 6+6^2\\\\\text{It is not in the form of}~ A^2-2AB+B^2[/tex]
At a sale, a desk is being sold for 29% of the regular price. The sale price is $272.60.
What is the regular price?
Answer:$940
Step-by-step explanation:
29%=$272.60
100%=
100%*$272.60/29%=$940
$940
Megan purchased a used car for $6,200. Her state charges 6% tax for the car, $57 for registration, $60 for a new title certificate, and $40 for a state safety and emissions inspection. Show all work. a. How much does Samantha need to pay for these extra charges, not including the price of the car? b. How much does she need to pay in total, for the car and extra charges? Show oyur work.
If A:B = 7:8 an B:C =12:7,find
HELP
explain what the answer is
Answer:10/7 and mixed number form is 1 3/7
Find the Perimeter of the Square with One side is 75cm
Answer:
P=300cm
Step-by-step explanation:
P=4a=4.75×4=300cm
[tex]\green{ \underline { \boxed{ \sf{Perimeter \: of \:Square = 4 \times side }}}}[/tex]
Putting Values -
[tex]\begin{gathered}\\\implies\quad \sf Perimeter \:of \:Square = 4 \times 75 \\\end{gathered} [/tex]
[tex]\begin{gathered}\\\implies\quad \sf Perimeter \: of \:Square = 300 sq. \:cm \\\end{gathered} [/tex]
>>Therefore, Perimeter of the Square is 300 sq. cm
What is the value of the expression 25 ÷ 5 + (6 x 2) − 4?
7
9
12
13
Answer: 13
Step-by-step explanation:
Answer:
Hey Dude....
Step-by-step explanation:
This is ur answer.....
25 ÷ 5 + (6 x 2) − 4= 5 + 12 - 4= 5 + 8 = 13Hope it helps!
Brainliest pls!
Follow me! :)
write 2.36 x 10^-3 as a decimal number (ordinary form)
Answer:
0.00236
Step-by-step explanation:
Move the decimal left 3 times as the exponent is negitiave 3
3 8 B. Maria bought 13 pounds of nails for a carpentry project. She ended up with pound of nails when she finished the project. a. Write an equation that can be used to determine how much of the 13 pounds of nails Maria used
I'm looking for a equation.
Step-by-step explanation:
let 1 pound of nail = x
so, the carved out waste nail will be = 12x
total nails = 13x
so the equation is =
13x - 12x = x
OR
x + 12x = 13x
Or
13x - x = 12x
Please Help ASAP (show work if you want its not needed i just need the answer)
[tex] \sin(60) = \frac{a}{4 \sqrt{3} } \\ [/tex]
[tex] \frac{ \sqrt{3} }{2} = \frac{a}{4 \sqrt{3} } \\ [/tex]
[tex]a = 6[/tex]
_____________________________________________
[tex] \cos(60) = \frac{c}{4 \sqrt{3} } \\ [/tex]
[tex] \frac{1}{2} = \frac{c}{4 \sqrt{3} } \\ [/tex]
[tex]c = 2 \sqrt{3} [/tex]
_____________________________________________
[tex] \sin(45) = \frac{a}{b} \\ [/tex]
[tex] \frac{ \sqrt{2} }{2} = \frac{6}{b } \\ [/tex]
[tex]b = 6 \sqrt{2} [/tex]
_____________________________________________
[tex]d = a = 6[/tex]
Have a great day ♡
7. Assessment Practice Tiffany buys a stuffed animal for 84¢. She pays with 8 dimes and I nickel. Which shows hov much change Tiffany should get? A A C) BO D
Answer:
1 cent
Step-by-step explanation:
Dime = 10 cents
8 Dimes = 8 * 10 = 80 Cents
1 Nickel = 5 Cents
80 + 5 = 85
85 - 84 = 1 Cent
solve this sum plsss urgent
1. 1 and -1 are two such rational numbers whose multiplicative inverse is same as they are.
2. absolute value is resembled using the formula : |x|
Here,
[tex]\hookrightarrow \sf |\dfrac{-3}{11} |[/tex]
[tex]\hookrightarrow \sf \dfrac{3}{11}[/tex]
final answer: 3/11
3.
[tex]\hookrightarrow \sf \dfrac{1}{2} \div \dfrac{3}{5}[/tex]
[tex]\hookrightarrow \sf \dfrac{1}{2} * \dfrac{5}{3}[/tex]
[tex]\hookrightarrow \sf \dfrac{5}{6}[/tex]
4.
Associative property can only be used with addition and multiplication
If you deposit $3,000 in a savings account that earns simple interest at a rate of 2.5% a year, how much interest would you have after 4 years?
Work Shown:
i = P*r*t
i = 3000*0.025*4
i = 300
A triangular prism and its dimensions are shown in the diagram. What is the lateral surface area of this triangular prism in square feet?
Answer:
112 ft
Step-by-step explanation:
AL = (a+b+c)h
= (6+12+10) x 4
= 112 ft
In the diagram, mZXYZ (Exterior Angle Y)=115º and
mZWXY (Angle X) = 45°.
What is m
Answer: 70 degrees
Step-by-step explanation:
Consider the equation 21 = m + 14.1. Which step will help to solve this equation? = a) Add 14 14 on both sides of the equation. . b) Subtract 14 tão on both sides of the equation. С Multiply both sides of the equation by 14 141 . o Divide both sides of the equation by 14
Answer:
B is the right answer
Step-by-step explanation:
Subtract 14 from each side to get the value of m
21 - 14 = 7 = m
m = 7
All exponential functions can be written in many forms. Write the function
f(t) = 2800(0.95)' in the form f(t) = abro. Round all coefficients to four
decimal places.
[tex]2800(0.95)^t~~ = ~~2800(b)^{\frac{t}{10}}\implies \cfrac{2800(0.95)^t}{2800}=b^{\frac{t}{10}}\implies 0.95^t=b^{\frac{t}{10}} \\\\\\ (0.95)^{\frac{10t}{10}}=b^{\frac{t}{10}}\implies (0.95^{10})^{\frac{t}{10}}=b^{\frac{t}{10}}\implies (0.5987)^{\frac{t}{10}}\approx b^{\frac{t}{10}}\implies 0.5987 \approx b \\\\[-0.35em] ~\dotfill\\\\ ~\hfill f(t)\approx 2800(0.5987)^{\frac{t}{10}}~\hfill[/tex]
Señora Cruz will use four triangles on the door decor. How many square centimeters of paper will Señora Cruz use to create the triangles?
Triangle has bases as 3.7 and height as 6.8
Square has length as 7.5 and width as 3.9
Answer:
50.32 cm²
Step-by-step explanation:
The area of a triangle can be computed using the formula ...
A = 1/2bh
For a triangle with base 3.7 cm and height 6.8 cm, the area is ...
A = 1/2(3.7 cm)(6.8 cm) = 12.58 cm²
__
For four (4) triangles, Señora Cruz will need 4 times this area:
4 × (12.58 cm²) = 50.32 cm²
Señora Cruz will need 50.32 cm² of paper to create her door decor.
There are 40%bananas and 50% apples in a bowl. How many more percentage points apples should be there so that apples become 50% more than bananas
Answer:
This means 50% of the total should be apples regardless of how many fruits there are in the bowl. If the reader only has 40% apples, they know they need to increase the number of...
Step-by-step explanation:
PLEASE ANSWER FAST WILL MARK BRAINLIEST
[tex]\sf y=50\left(1.8\right)^{x}[/tex]
=================================
the first term : 50
crosses point : (0, 50), (1, 90)
solve for y,
[tex]\sf y=50\left(1.8\right)^{1}[/tex]
[tex]\sf y=90[/tex]
Thus D is correct option.Graph:
ans this sum plsss .
Answer:
See below ↓
Step-by-step explanation:
3.
2/512/253/73/83/104.
Draw a number of spacing 0.25 eachStart from -1 to 1-2/8 = -0.25 [mark on line]3/6 = 0.5 [mark on line]5.
Midpoint of -2/5 and 1/2M = [-2/5 + 1/2] / 2M = [1/10] / 2M = 1/20I need help with this question
Answer:
301 feet
Step-by-step explanation:
plug in the known values
[tex]85=\sqrt{24d}[/tex]
square both sides to cancel the square root
[tex]7225=24d[/tex]
divide both sides by 24 to get d on its own
[tex]d=301.041[/tex] feet
to the nearest foot is 301 feet
Factor completely 2x2 – X – 4.
(3x - 1)(x + 4)
(3x + 4)(x - 1)
(3x − 2)(x + 2)
-
(3x – 4)(x + 1)
-
[tex]2x {}^{2} - x - 4[/tex]
[tex]x = \frac{ - ( - 1) + \sqrt{1 - 4(2)( - 4)} }{2(2)} = \frac{1 + \sqrt{33} }{4} [/tex]
[tex]x = \frac{1 - \sqrt{33} }{4} [/tex]
[tex](x - \frac{1 + \sqrt{33} }{4} )(x - \frac{1 - \sqrt{33} }{4} )[/tex]
[tex](4x - 1 - \sqrt{33} )(4x - 1 + \sqrt{33} )[/tex]
It takes _____________ for the minute hand to travel from one number to the next.
A) 10 minutes
B) 15 minutes
C) 1 minute
D) 5 minutes
Answer:
D)
Step-by-step explanation:
Because -12 to 1 is 5 min.
Answer:
it takes five minutes for the minute hand to travel from one number to the next
Step-by-step explanation:
5
[tex]\underline{\boxed{\blue{\Large{\bf{Challenge}}}}}[/tex]
If 'P and 'Q' are two points whose coordinates are (at^2, 2at) and (a/t^2, 2a/t) respectively and S is the point(0, 0). Show that 1/SP + 1/SQ is independent of 't'.
Note:-
Plagarised/spam/short answers will be deleted on the spot.
Answer with all steps and proper explanation .
Step-by-step explanation:
Given point are: P(at², 2at)
Q(a/t², 2a/t)
S(0, 0)
Now, the distance between A(x₁y₁) and B(X₂, y₂)
then AB = √{(x₂ - x₁)² + (y₂ - y₁)²} units
(i) The distance between S and P:
(x₁, y₁) = (0, 0) ⇛x₁= 0, y₁ = 0
(x₂, y₂ ) = (at², 2at) ⇛x₂ = at², y₂ = 2at
SP = √{at² - 0)² + (2at - 0)²}
= √{(at²)² + (2at)²}
= √{a²t²*² + 4a²t²}
= √{a²t⁴ + 4a²t²}
= √{a²t²(t²+4)}
SP = at√(t² + 4)→→→Eqn(1)
(ii) The distance between S and Q :
(x₁, y₁) = (0, 0) ⇛x₁= 0, y₁ = 0
(x₂, y₂ ) = (a/t², 2a/t) ⇛x₂ = a/t², y₂ = 2a/t
SQ = √[{(a/t²) - 0} + {(2a/t) - 0}
= √{(a/t²)² + (2a/t)²}
= √{(a²/t²*²) + (4a²/t²)}
= √{(a²/t⁴) + (4a²/t²)}
= √{(a² + 4a²t²)/t⁴}
= √[{a²(1 + 4t²)}/t⁴]
SQ = (a/t²)√(1 + 4t²) →→→ Eqn(2)
Now,
(1/SP) + (1/SQ) = [1/{at√(t² + 4)}] + [1/{(a/t²)√(1 + 4t²)}]
= (1/at)[1/{√(t² + 4)}] + (t²/a)[1/{(√1 + 4t²)}]
= (1/a)[[1/{t√(t² + 4)}] + [t²/{√(1 + 4t²)}]]
(1/SP) + (1/SQ) = 1/a is not independent of 't'
If suppose S = (a, 0) then
SP = √{(at² - a)² + (2at - 0)²}
= √{a²(t² - 1)² + (2at)²}
= a√{(t² - 1)² + 4t²}
= a√{(t² + 1)²}
SP = a(t² + 1)
1/SP = 1/{a(t² + 1)} →→→Eqn(1)
And
SQ = √[{(a/t²) - a}² + {(2a/t) - 0}²]
= √[a²{(1/t²) - 1}² + a²(2/t)²]
= a√[{(1 - t²)²/t⁴} + (4/t²)]
= a√[{(1 - t²)² + 4t²}/t⁴]
( a/t²)√(1 + t²)²
SQ = (a/t²)(1 + t²)
1/SQ = 1/{(a/t²)(1 + t²)} = t²/{a(1 + t²)} →→→Eqn(2)
Therefore, (1/SP) + (1/SQ)
= 1/{a(t² + 1)} + t²/{a(1 + t²)}
= (1 + t²)/a(1 + t²)
= 1/a
1/a is independent of 't'.
PLEASE HELP PLEASE HELP
Answer:
Acute angle
Step-by-step explanation:
Acute angle = less than 90 degrees
Answer:
acute angles
Step-by-step explanation:
Angles between 0 and 90 degrees (0°< θ <90°) are called acute angles. Angles between 90 and 180 degrees (90°< θ <180°) are known as obtuse angles. Angles that are 90 degrees (θ = 90°) are right angles.
Solve for d
Can you pls help me?
Find an equation of the vertical line through (-4,-3)
Answer:
x = -4
Step-by-step explanation:
Vertical lines are up and down, which means the x coordinate stays the same.
This means the line is in the form
x=
x = -4
Answer:
Step-by-step explanation:
A vertical line has the general equation of x = a. y can be anything.
The x value that you will use is -4. Every point must have -4 as its value for a.
Answer: x = -4
What inverse operation will be used to undo the volume of Julie's jewelry box to find its edge lengths?
Volume formula for a cube is: volume = edge ^ 3
Answer:
Cubed root
Step-by-step explanation:
Do the inverse of cubed, cubed root
The mean of 5 different numbers is 6 and the mean of 4 different
numbers is 15. Find the mean of 9 numbers. Please answer fast as possible.