Mathematics High School

Hello ! Here's a calculus question!

Find that value of

[tex]\\ \rm\Rrightarrow {\displaystyle{\int\limits_{\pi}^{2\pi}}}\dfrac{sin^6x+cos^6x}{sin^3xcos^3x}[/tex]


Note:-

Answer must include all steps properly .

Kindly don't waste time here if you don't know the answer .

All the best !​

Answers

Answer 1

Answer:

Undefined.

Formula's used:

[tex]\longrightarrow \bold{sec(x) = \dfrac{1}{cos(x)} }[/tex]

[tex]\longrightarrow \bold{cosec(x) = \dfrac{1}{sin(x)} }[/tex]

[tex]\longrightarrow \bold{cot(x) = \dfrac{1}{tan(x)} }[/tex]

[tex]\longrightarrow \bold{tan(x) = \dfrac{sinx}{cos(x)} }[/tex]

[tex]\longrightarrow \bold{sin^2x + cos^2 x = 1 }[/tex]

[tex]\longrightarrow \sf \bold{Sum \ Rule}:\quad \int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx}[/tex]

[tex]\longrightarrow \bold{ \int \:{sec^2(ax+b) = \dfrac{1}{a}tan(ax+b)+c }}[/tex]

[tex]\longrightarrow \bold{\int \:\dfrac{1}{ax+b} =\dfrac{1}{a} ln|ax+b|+c}[/tex]

Explanation:

[tex]\sf \Longrightarrow \int _{\pi }^{2\pi }\:\dfrac{sin^6(x)+cos^6(x)}{sin^3(x) \ * \ cos^3\left(x)}[/tex]

[tex]\sf \Longrightarrow \sf \int _{\pi }^{2\pi }\:\dfrac{sin^6\left(x\right)}{sin^3\left(x\right)\cdot \:cos^3\left(x\right)} +\dfrac{cos^6\left(x\right)}{sin^3\left(x\right)\cdot \:cos^3\left(x\right)}[/tex]

[tex]\Longrightarrow \sf \int _{\pi }^{2\pi }\:\dfrac{sin^3\left(x\right)}{\:cos^3\left(x\right)} +\dfrac{cos^3\left(x\right)}{sin^3\left(x\right)}[/tex]

                                                               

[tex]\Longrightarrow \sf \int _{\pi }^{2\pi }\ tan^3(x)+ cot^3(x)[/tex]

[tex]\sf \Longrightarrow \sf \int _{\pi }^{2\pi } \tan ^3(x)dx+\int _{\pi }^{2\pi } \cot ^3 (x)dx[/tex]

[tex]\Longrightarrow \sf \bold{ [ }-\ln |\sec \left(x\right) |+\dfrac{\sec ^2(x)}{2}-\dfrac{\cot ^2 (x)}{2}-\ln |\sin(x)| \bold{ ] }^{2\pi }_\pi[/tex]

apply limits

[tex]\sf \Longrightarrow \sf -\ln \left|\sec \left(2\pi \right)\right|+\dfrac{\sec ^2\left(2\pi\right)}{2}-\dfrac{\cot ^2\left(2\pi\right)}{2}-\ln \left|\sin \left(2\pi\right)\right|-(-\ln \left|\sec \left(\pi \right)\right|+\dfrac{\sec ^2\left(\pi\right)}{2}-\dfrac{\cot ^2\left(\pi\right)}{2}-\ln \left|\sin \left(\pi\right)\right|)[/tex]

simplify using trigonometric basic functions

[tex]\Longrightarrow \sf -\ln \left|\dfrac{1}{\cos \left(2\pi \right)}\right|+\dfrac{\left(\dfrac{1}{\cos \left(2\pi \right)}\right)^2}{2}-\dfrac{\cot ^2\left(2\pi \right)}{2}-\ln \left|\sin \left(2\pi \right)\right|- ( -\ln \left|\dfrac{1}{\cos \left(\pi \right)}\right|+\dfrac{\left(\frac{1}{\cos \left(\pi \right)}\right)^2}{2}-\dfrac{\cot ^2\left(\pi \right)}{2}-\ln \left|\sin \left(\pi \right)\right|)[/tex]

The value of cot(2π) is not defined. [ cot(2π) = ∞ ]

⇒  Undefined

Answer 2

Answer:

[tex]\displaystyle \int\limits^{2 \pi}_{\pi} {\frac{\sin^6 x + \cos^6 x}{\sin^3 x \cos^3 x}} \, dx = \boxed{\text{un} \text{de} \text{f}}[/tex]

General Formulas and Concepts:
Calculus

Differentiation

DerivativesDerivative Notation

Integration

Integrals

Integration Rule [Reverse Power Rule]:
[tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]

Integration Rule [Fundamental Theorem of Calculus 1]:
[tex]\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)[/tex]

Integration Property [Multiplied Constant]:
[tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]

Integration Property [Addition/Subtraction]:
[tex]\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx[/tex]

Integration Methods: U-Substitution + U-Solve

Step-by-step explanation:

Step 1: Define

Identify given.

[tex]\displaystyle \int\limits^{2 \pi}_{\pi} {\frac{\sin^6 x + \cos^6 x}{\sin^3 x \cos^3 x}} \, dx[/tex]

Step 2: Integrate Pt. 1

[Integrand] Rewrite:
[tex]\displaystyle \frac{\sin^6 x + \cos^6 x}{\sin^3 x \cos^3 x} = \frac{\sin^6 x}{\sin^3 x \cos^3 x} + \frac{\cos^6 x}{\sin^3 x \cos^3 x}[/tex][Integrand] Simplify:
[tex]\displaystyle \frac{\sin^6 x + \cos^6 x}{\sin^3 x \cos^3 x} = \tan^3 x + \cot^3 x[/tex][Integrand] Rewrite:
[tex]\displaystyle \frac{\sin^6 x + \cos^6 x}{\sin^3 x \cos^3 x} = \tan x (\sec^2 x - 1) + \cot x (\csc^2 x - 1)[/tex]


Step 3: Integrate Pt. 2

[Integral] Rewrite:
[tex]\displaystyle \int\limits^{2 \pi}_{\pi} {\frac{\sin^6 x + \cos^6 x}{\sin^3 x \cos^3 x}} \, dx = \int\limits^{2 \pi}_{\pi} {\tan x (\sec^2 x - 1) + \cot x (\csc^2 x - 1)} \, dx[/tex][Integral] Rewrite [Integration Property - Addition/Subtraction]:
[tex]\displaystyle \int\limits^{2 \pi}_{\pi} {\frac{\sin^6 x + \cos^6 x}{\sin^3 x \cos^3 x}} \, dx = \int\limits^{2 \pi}_{\pi} {\tan x (\sec^2 x - 1)} \, dx + \int\limits^{2 \pi}_{\pi} {\cot x (\csc^2 x - 1)} \, dx[/tex]

Step 4: Integrate Pt. 3

Identify variables for u-solve.

1st Integral

Set u:
[tex]\displaystyle u = \sec x[/tex][u] Apply Trigonometric Differentiation:
[tex]\displaystyle du = \sec x \tan x \, dx[/tex][du] Rewrite:
[tex]\displaystyle dx = \frac{1}{\sec x \tan x} \, du[/tex]

2nd Integral

Set v:
[tex]\displaystyle v = \csc x[/tex][v] Apply Trigonometric Differentiation:
[tex]\displaystyle dv = - \cot x \csc x \, dx[/tex][dv] Rewrite:
[tex]\displaystyle dx = \frac{-1}{\cot x \csc x} \, dv[/tex]

Step 5: Integrate Pt. 4

[Integrals] Apply Integration Method [U-Solve]:
[tex]\displaystyle \int\limits^{2 \pi}_{\pi} {\frac{\sin^6 x + \cos^6 x}{\sin^3 x \cos^3 x}} \, dx = \int\limits^{x = 2 \pi}_{x = \pi} {\frac{\tan x (\sec^2 x - 1)}{\sec x \tan x}} \, du + \int\limits^{x = 2 \pi}_{x = \pi} {\frac{- \cot x (\csc^2 x - 1)}{\cot x \csc x}} \, dv[/tex][Integrals] Simplify:
[tex]\displaystyle \int\limits^{2 \pi}_{\pi} {\frac{\sin^6 x + \cos^6 x}{\sin^3 x \cos^3 x}} \, dx = \int\limits^{x = 2 \pi}_{x = \pi} {\frac{u^2 - 1}{u}} \, du - \int\limits^{x = 2 \pi}_{x = \pi} {\frac{v^2 - 1}{v}} \, dv[/tex][Integrals] Apply Integration Rule [Reverse Power Rule]:
[tex]\displaystyle \int\limits^{2 \pi}_{\pi} {\frac{\sin^6 x + \cos^6 x}{\sin^3 x \cos^3 x}} \, dx = \frac{u^2}{2} \bigg| \limits^{x = 2 \pi}_{x = \pi} - \int\limits^{x = 2 \pi}_{x = \pi} {\frac{1}{u}} \, du - \Bigg( \frac{v^2}{2} \bigg| \limits^{x = 2 \pi}_{x = \pi} - \int\limits^{x = 2 \pi}_{x = \pi} {\frac{1}{v}} \, dv \Bigg)[/tex][Integrals] Apply Logarithmic Integration:
[tex]\displaystyle \int\limits^{2 \pi}_{\pi} {\frac{\sin^6 x + \cos^6 x}{\sin^3 x \cos^3 x}} \, dx = \frac{u^2}{2} \bigg| \limits^{x = 2 \pi}_{x = \pi} - \ln | u | \bigg| \limits^{x = 2 \pi}_{x = \pi} - \Bigg( \frac{v^2}{2} \bigg| \limits^{x = 2 \pi}_{x = \pi} - \ln | v | \bigg| \limits^{x = 2 \pi}_{x = \pi} \Bigg)[/tex]Back-Substitute variables u and v:
[tex]\displaystyle \int\limits^{2 \pi}_{\pi} {\frac{\sin^6 x + \cos^6 x}{\sin^3 x \cos^3 x}} \, dx = \frac{\sec^2 x}{2} \bigg| \limits^{2 \pi}_{\pi} - \ln | \sec x | \bigg| \limits^{2 \pi}_{\pi} - \Bigg( \frac{\csc^2 x}{2} \bigg| \limits^{2 \pi}_{\pi} - \ln | \csc x | \bigg| \limits^{2 \pi}_{\pi} \Bigg)[/tex]Simplify:
[tex]\displaystyle \int\limits^{2 \pi}_{\pi} {\frac{\sin^6 x + \cos^6 x}{\sin^3 x \cos^3 x}} \, dx = \bigg( \ln | \csc x | - \ln | \sec x | + \frac{\sec^2 x - \csc^2 x}{2} \bigg) \bigg| \limits^{2 \pi}_{\pi}[/tex]Apply Integration Rule [Fundamental Theorem of Calculus 1]:
[tex]\displaystyle \int\limits^{2 \pi}_{\pi} {\frac{\sin^6 x + \cos^6 x}{\sin^3 x \cos^3 x}} \, dx = \boxed{\text{un} \text{de} \text{f}}[/tex]

∴ we have found the value of the given integral.

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Learn more about integration: https://brainly.com/question/14413972

Learn more about calculus: https://brainly.com/question/20197752

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Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Integration

Related Questions

in a class children, three quarters are chinese, one fifth are malay, and the rest are indian. what fraction of the class are indian?

Answers

Answer:

[tex] \frac{1}{20} [/tex]

or 5%

Step-by-step explanation:

Explanation:

3/4=15/20

1/5=4/20

So, rest of 'em 1/20

Answer:

1/20.

Step-by-step explanation:

1 - 3/4 - 1/5

= 20/20  - 15/20 - 4/20

= 1/20

6. The table below represents the number of
donut holes per box at Damon's Donuts. What
is the median number of donut holes?
# OF DONUT
HOLES
22
23
24
25
FREQUENCY
1
2
3
2
A. 23
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Answers

C) 24.

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After she finished making the cookies, Ashley had 0.945 kg of flour lett. How much flour did she use?​

Answers

Answer:

[tex]she \: used \: 1.695 \: kg \: of \: flour.[/tex]

Step-by-step explanation:

[tex]2.64 - 0.945 = 1.695[/tex]

Obviously the question is not complete but if it’s that type of question then she used
x minus 0.945 kg.
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Answers

Answer:

f(g))(0) = 0

Step-by-step explanation:

assuming you mean f(g)(0)

evaluate g(0) then substitute the value obtained into f

x = 0 then g(0) = 2 ← from graph (0, 2 ) , then

x = 2 then f(2) = 0 ← from graph (2, 0 )

Find the length of a path that runs diagonally across a 20-yard by 50-yard field

Answers

Answer:

c = 53.85 or 54

Step-by-step explanation:

Which of the following are examples of exponential decay? Select all that apply.
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A vehicle depreciates 13% each year.
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D
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A retirement account decreases by $12,000 each year.
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A retirement account decreases by $12,000 each year

Use the picture I provided to help me answer the question.

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B


Cuz there is 9 flutes and 4 violas and 9 is greater than 4

PLEASE HELP ME I REALLY NEED IT

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Answer: 170

Step-by-step explanation: Angle G and Angle H are consecutive angles, meaning they both add up to 180 degrees. Thus, if 10 is the value of Angle G, Angle H must be 170.

on the number line,label the points that are 5 over 2 units away from 0. What is the distance between these points

Answers

Answer: 5 units

Step-by-step explanation:

5/2 = 2.5

they are 2.5 away on each side of 0 therefore the points are 5 units away from each other

Consider the following polynomial

Answers

Step-by-step explanation:

Degree is 6

Leading coefficient is 5

Constant term is 8

i need help 9th grade math

Answers

Answer:

-x + 1

Step-by-step explanation:

f(x) = 2x + 3

g(x) = 3x + 2

we want to find (f-g)(x)

(f-g)(x) is the same as saying f(x) - g(x)

f(x) - g(x) = (2x + 3) - (3x + 2)

==> remove parenthesis and apply negative sign

f(x) - g(x) = 2x + 3 - 3x - 2

==> combine like terms

f(x) - g(x) = -x + 1

(f-g)(x)f(x)-g(x)

Put values

2x+3-(3x+2)2x+3-3x-2-x+1

what is the slope of (2,5) (-6,-3)

Answers

Answer:

m=1

Step-by-step explanation:

(2,5);(−6,−3)

(x1,y1)=(2,5)

(x2,y2)=(−6,−3)

Use the slope formula:

m=

y^2−y^1

x^2−x^1

=

−3−5/−6−2

=

−8/−8

=1

Answer:

m=1

Answer:

[tex]\boxed{\sf{1}}[/tex]

Step-by-step explanation:

To solve this problem, you have to use a slope formula.

Slope formula:

[tex]\Longrightarrow: \sf{\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{Rise}{run} }[/tex]

[tex]\sf{y_2=(-3)}\\\\\sf{y_1=5}\\\\\sf{x_2=(-6)}\\\\\sf{x_1=2}[/tex]

Rewrite the problem down.

[tex]\sf{\dfrac{(-3)-5}{(-6)-2} }[/tex]

Solve.

[tex]\sf{\dfrac{(-3)-5}{(-6)-2}=\dfrac{-8}{-8}=\boxed{\sf{1} }[/tex]

Therefore, the slope is 1, which is our answer.

I hope this helps you! Let me know if my answer is wrong or not.

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Answers

Answer:

•construction of buildings

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Answers

To find the area of an irregular shape (in general)

 ⇒ is best to split it into different regular shapes

   ⇒ (in this case), let's split into a half-circle and a square

Area of the half-circle:

 ⇒ half the area of a circle ([tex]\pi r^2[/tex])

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r: length of radius ⇒ half of length of the diameter ⇒ 8.5m[tex]\pi[/tex] = 3.14

    Area of Half Circle = [tex]\frac{1}{2}\pi (8.5)^2 = 113.5m^2[/tex]

Area of square

  ⇒ length * width

     ⇒ Area of Square = 17 * 17 = 289[tex]m^2[/tex]

Total Area = 113.5 + 289 = 402.5 ≈ 402[tex]m^2[/tex]

Hope that helps!

 

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Answers

Answer:

C

Step-by-step explanation:

Since the angles in a triangle add to 180 degrees, the third angle is 180-70-45=65 degrees

There were 15 birds in the park. Some (n) of them flew away. Which expression could be
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Answers

Answer:

15-n

Step-by-step explanation:

subtract the number that flew away from the total number of birds that were there

Which of the following statements is no True?
A) the square root of 112 is between 10 and 11
B) the square root of 180 is between 13 and 14
C) the square root of 12 is between 4 and 5
D) the square root of 5 is between 2 and 3

Answers

Answer:

correct answer is A. because the square root of 122 is 11.06 which means its between 11 and 12, not 10 and 11

Step-by-step explanation:

A farmer has an equal number of horses in 4 different fields on his farm. Which choice could be the total
number of horses?

205
232
253
289

Answers

Answer:

B -- 232

Step-by-step explanation:

because you can divide 232 by 4 and get a whole number.

The diameter of a circle is 18 ft. Find its area in terms of pi

Answers

Answer:

81π

Step-by-step explanation:

To find the area of a circle, the formula is πr^2

r=radius, and radius is half the diameter, so it is 9.

Plug the radius into the formula to get 81π

(9^2*π)

If this answer helped you, I would greatly appreciate the brainliest :D

which is correct and explain

Answers

Answer:

c

Step-by-step explanation:

Answer 1-30 With work shown Please and Thankyou

Answers

Answer:

Find the following square and cube roots.

1. √144 2.√16 3.√289

= 12 = 4 = 17

4. √(4/81) 5. √25 6. √400

= 0.222 = 5 =20

7. 3√27 8. 3√-8 9. 3√216

= 15.588 = -2 = 6

10. 3√343 11. 3√-1000 12. 3√1

= 7 = -10 = 3

________________________________

Simplify the following square and cube roots. SHOW WORK!

13. √112

= solution,

Cube roots, square roots,

3√112 2√112

= 3√(2x2x2)x2x7 = 4√28

= 2 ∛14 = 4√7

= 4.82

14. √20

=

Find the area of a regular octagon with an apothem length of 10 and a side length of 6.

Answers

Answer:

A = 240 units²

Step-by-step explanation:

the area (A) of a regular polygon is

A = [tex]\frac{1}{2}[/tex] pa ( p is the perimeter and a the apothem )

here p = 8 × 6 = 48 ( an octagon has 8 sides ) and a = 10 , then

A = [tex]\frac{1}{2}[/tex] × 48 × 10 = 24 × 10 = 240 units²

Answer:

Step-by-step explanation:

Preliminary task

Draw a regular octagon.

Draw a perpendicular from the side's midpoint to the center. (That is the  apothem).

Draw two radii from the center of the side to the center of of the octagon. The endpoints of these two radii is the endpoints of the octagon's side.

The octagon has 8 such triangles.

Area one triangle

Area = 1/2 * side *  apothem

side = 6

apothem = 10

Area = 1/2 6 * 10

Area = 30

Area of 8 such triangles.

Area = 8 * area of 1 triangle

area 1 triangle = 30

Area of 8 such triangles = 30*8 = 240

Answer 240

what is the value of x?

6(x-4)=10

Answers

Answer:

[tex]\boxed{\sf{x=\dfrac{17}{3}=5.6 }}[/tex]

Step-by-step explanation:

Use a distributive property.

Distributive property:

[tex]\Longrightarrow: \sf{A(B+C)=AB+AC}[/tex]

⇒ 6(x-4)=10

First, divide by 6 from both sides.

⇒ 6(x-4)/6=10/6

Solve.

⇒ x-4=5/3

Add by 4 from both sides.

⇒ x-4+4=5/3+4

Solve.

5/3+4

4*3/3+5/3

4*3+5/3

4*3=12

12+5=17

x=17/3

Dividing is another option.

17/3=5.6

x=17/3=5.6

Therefore, the final answer is x=17=5.6.

I hope this helps you! Let me know if my answer is wrong or not.

i really need help please

Answers

Answer:

56⁰

Step-by-step explanation:

I hope it helps you

If any doubts do let me know in the comments

WILL GIVE BRAINLIEST

A student was calculating the residuals of a linear regression equation. The largest residual value found was between 0.90 and 1. What could be said about the data?

Answers

Answer:there is a very strong corolation

Step-by-step explanation:

Find the mean and the median of this data: 9, 6, 5, 3, 28, 6, 4, 7.

Answers

Answer: 9 is mean and median is 6

Step-by-step explanation:

Ill give you a cookie if you answer the question correct

Answers

Answer:

$7.20

Step-by-step explanation:

convert 1 1/4 to decimal form.

0.25+1=1.25

9/1.25 =7.20

If f(1)=2 and f(n) = f(n-1)^2+4 then what is the value of f(3)?

Answers

Answer:

f(3) = 68

Step-by-step explanation:

Given:

[tex]\sf f(1)=2[/tex]

[tex]\sf f(n) = f(n-1)^2+4[/tex]

[tex]\sf \implies f(2) = f(1)^2+4[/tex]

[tex]\sf \implies f(2) = 2^2+4=8[/tex]

[tex]\sf \implies f(3) = f(2)^2+4[/tex]

[tex]\sf \implies f(3) = 8^2+4=68[/tex]

Which function is graphed below?

Answers

Answer:

The function y =f (x) is graphed below. Plot a line segment connecting the points on f where x = -9 and x = -3. Use the line segment to determine the average rate of change of the function f (x) on the interval -9 <x <-3?

Step-by-step explanation:

I am not sure where or what you are talking about?

Which of the following expressions are equivalent to -2(5 – 3)?
Choose all answers that apply:
A
-10 - 6
B
-10 + 6
None of the above
Peno
Sos

Answers

Answer:

It's B

-10 + 6

Step-by-step explanation:

Here, we gotta use the distributive property.

the -2 will distribute to the 5, which will make the -10

then, the -2 will also distribute to the -3 (the minus is the negative) which will make the 6

So, that should give you the -10 + 6, which is B

hope this helped :)

Answer:

[tex]\boxed{\sf{B.-10+6}}[/tex]

Step-by-step explanation:

Use the distributive property.

Distributive property:

→ A(B+C)=AB+AC

→ -2(5-3)

Multiply.

→ -2*5=-10

→ -2*3=-6

Rewrite the problem down.

→ = -10+6

Therefore, the correct answer is B. -10+6.

I hope this helps you! Let me know if my answer is wrong or not.

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