Replace [tex]x\mapsto \tan^{-1}(x)[/tex] :
[tex]\displaystyle \int_0^{\frac\pi2} \sqrt[3]{\tan(x)} \ln(\tan(x)) \, dx = \int_0^\infty \frac{\sqrt[3]{x} \ln(x)}{1+x^2} \, dx[/tex]
Split the integral at x = 1, and consider the latter one over [1, ∞) in which we replace [tex]x\mapsto\frac1x[/tex] :
[tex]\displaystyle \int_1^\infty \frac{\sqrt[3]{x} \ln(x)}{1+x^2} \, dx = \int_0^1 \frac{\ln\left(\frac1x\right)}{\sqrt[3]{x} \left(1+\frac1{x^2}\right)} \frac{dx}{x^2} = - \int_0^1 \frac{\ln(x)}{\sqrt[3]{x} (1+x^2)} \, dx[/tex]
Then the original integral is equivalent to
[tex]\displaystyle \int_0^1 \frac{\ln(x)}{1+x^2} \left(\sqrt[3]{x} - \frac1{\sqrt[3]{x}}\right) \, dx[/tex]
Recall that for |x| < 1,
[tex]\displaystyle \sum_{n=0}^\infty x^n = \frac1{1-x}[/tex]
so that we can expand the integrand, then interchange the sum and integral to get
[tex]\displaystyle \sum_{n=0}^\infty (-1)^n \int_0^1 \left(x^{2n+\frac13} - x^{2n-\frac13}\right) \ln(x) \, dx[/tex]
Integrate by parts, with
[tex]u = \ln(x) \implies du = \dfrac{dx}x[/tex]
[tex]du = \left(x^{2n+\frac13} - x^{2n-\frac13}\right) \, dx \implies u = \dfrac{x^{2n+\frac43}}{2n+\frac43} - \dfrac{x^{2n+\frac23}}{2n+\frac23}[/tex]
[tex]\implies \displaystyle \sum_{n=0}^\infty (-1)^{n+1} \int_0^1 \left(\dfrac{x^{2n+\frac43}}{2n+\frac43} - \dfrac{x^{2n+\frac13}}{2n-\frac13}\right) \, dx \\\\ = \sum_{n=0}^\infty (-1)^{n+1} \left(\frac1{\left(2n+\frac43\right)^2} - \frac1{\left(2n+\frac23\right)^2}\right) \\\\ = \frac94 \sum_{n=0}^\infty (-1)^{n+1} \left(\frac1{(3n+2)^2} - \frac1{(3n+1)^2}\right)[/tex]
Recall the Fourier series we used in an earlier question [27217075]; if [tex]f(x)=\left(x-\frac12\right)^2[/tex] where 0 ≤ x ≤ 1 is a periodic function, then
[tex]\displaystyle f(x) = \frac1{12} + \frac1{\pi^2} \sum_{n=1}^\infty \frac{\cos(2\pi n x)}{n^2}[/tex]
[tex]\implies \displaystyle f(x) = \frac1{12} + \frac1{\pi^2} \left(\sum_{n=0}^\infty \frac{\cos(2\pi(3n+1)x)}{(3n+1)^2} + \sum_{n=0}^\infty \frac{\cos(2\pi(3n+2)x)}{(3n+2)^2} + \sum_{n=1}^\infty \frac{\cos(2\pi(3n)x)}{(3n)^2}\right)[/tex]
[tex]\implies \displaystyle f(x) = \frac1{12} + \frac1{\pi^2} \left(\sum_{n=0}^\infty \frac{\cos(6\pi n x + 2\pi x)}{(3n+1)^2} + \sum_{n=0}^\infty \frac{\cos(6\pi n x + 4\pi x)}{(3n+2)^2} + \sum_{n=1}^\infty \frac{\cos(6\pi n x)}{(3n)^2}\right)[/tex]
Evaluate f and its Fourier expansion at x = 1/2 :
[tex]\displaystyle 0 = \frac1{12} + \frac1{\pi^2} \left(\sum_{n=0}^\infty \frac{(-1)^{n+1}}{(3n+1)^2} + \sum_{n=0}^\infty \frac{(-1)^n}{(3n+2)^2} + \sum_{n=1}^\infty \frac{(-1)^n}{(3n)^2}\right)[/tex]
[tex]\implies \displaystyle -\frac{\pi^2}{12} - \frac19 \underbrace{\sum_{n=1}^\infty \frac{(-1)^n}{n^2}}_{-\frac{\pi^2}{12}} = - \sum_{n=0}^\infty (-1)^{n+1} \left(\frac1{(3n+2)^2} - \frac1{(3n+1)^2}\right)[/tex]
[tex]\implies \displaystyle \sum_{n=0}^\infty (-1)^{n+1} \left(\frac1{(3n+2)^2} - \frac1{(3n+1)^2}\right) = \frac{2\pi^2}{27}[/tex]
So, we conclude that
[tex]\displaystyle \int_0^{\frac\pi2} \sqrt[3]{\tan(x)} \ln(\tan(x)) \, dx = \frac94 \times \frac{2\pi^2}{27} = \boxed{\frac{\pi^2}6}[/tex]
Find the perimeter of the figure.
6 meters
2y meters
2y
meters
4y meters
16 meters
4y meters
Answer:
34
Step-by-step explanation:
Usually when finding the perimeter all you have to do is add the sides.
What is x?
ONLY WRITE A NUMBER FOR YOUR ANSWER
If you need to round, round to the nearest tenth (one decimal place).
Please help, thank you!
Answer:
15
Step-by-step explanation:
to get from 10 to 25 you time by 2½
so then times 6 by 2½ and the answer is 15
Which is more, 1 pound or 22 ounces?
Answer:
22 ounces is more
Step-by-step explanation:
1 pound is equal to 16 ounces
Answer:
1 pound
Step-by-step explanation:
1pound=16oz 22oz= 1.375pound
what is 7/8 - 1/2 = in simplest form
Answer:
[tex] \frac{7}{8} - \frac{1}{2} = \frac{7 - 4}{8} = \frac{3}{8} [/tex]
Step-by-step explanation:
I hope I helped you! :)
Figure B is a scale image of Figure A, as shown. Enter the scale factor applied to Figure A to produce Figure B.
Having trouble with this, help please?
Answer:
a) [tex]7x-15=20[/tex]
b) [tex]\frac{20+15}{7}[/tex]
c) [tex]5[/tex]
Step-by-step explanation:
Find the area of the shaded shape. Write the area in square units
Answer:
length times width equals area LxW=Area
Step-by-step explanation:
so,
4 times 4= 26 yards
1=4 square yards
then 26X4 = 104 square yards
Answer:
Step-by-step explanation:
What is the exact value of cos(-60°)?
V3
2
Ni-
을
V3
2
Applying the cosine ratio of trigonometry, the exact value of cos(-60°) which equals cos (π/3) is: C. 1/2.
What is the Cosine Ratio?In trigonometry, the vaue of the cosine ratio is the cosine of the reference angle which equals the ratio of the adjcent side over the hypotenuse.
To get the exact value of cos(-60°), convert 60 to radians.
π radian = 180
60° = (π/3) radians
Note: cos (-θ) = cos θ
Therefore, cos (-π/3) = cos (π/3)
cos (π/3) = 1/2
The exact value of cos(-60°) = 1/2 (option C).
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Which prism has a greater volume?
Find the value of b,c, and d.
Answer:
b = 25 degrees
c = 81 degrees
d = 64 degrees
Which point is a point where the graph of [tex]y=(x-5)(x^2-7x+12)[/tex] crosses the x-axis?
Answer: C. (4, 0)
Step-by-step explanation:
We can graph the equation given to solve. See attached.
The answer to this question is C. (4, 0)
PLEASE HURRY!!!
A bag contains 4 red pencils, 6 blue pencils, and 5 green pencils. A pencil is randomly selected, its color is recorded, and it is returned to the bag. This is repeated 1350 times. How many times is it expected that a green pencil will be selected?
A. approximately 360 times
B. approximately 450 times
C. approximately 540 times
D. approximately 900 times
Answer:
450
Step-by-step explanation:
1. Based on the given conditions, formulate:
1350 x 5 ÷ (5 + 4 + 6)
2. Calculate
[tex]\frac{1350x5}{5 + 4 + 6}[/tex] The numerator is 1350 multiply by 5
[tex]\frac{6750}{5 + 4 + 6}[/tex]
[tex]\frac{6750}{15}[/tex]
Answer: 450
-y + 2y = what ? Please help and why is it negative or positive ?
Step-by-step explanation:
-y is the same as writing -1y. if we forget about the y’s, -1 + 2 = 1. so -1y + 2y = 1y, or just y to be simpler :)
i hope this helps :)
[tex]-y+2y=2y-y=(2-1)y=y[/tex]
It is positive because the coefficient is positive.
What is the equation in standard
form of a circle with the
equation x^2+ y^2 + 14y – 1 = 0?
Answer:
(x-2)²+(y+7)² = 25Step-by-step explanation:
The general standard form of an equation is written in expressed as shown;
(x-a)²+(y-b)² = r²
(a,b) is the centre of the circle and r is the radius
Given the equation of the circle x^2 + y^2 - 4x + 14y +28 =0, we need to find the centre of the circle and its radius.
comparing the equation given with x²+y²+2gx+2fy+c =0
(-g, -f) is the centre
radius = √g²+f²-c
On comparison,
2gx = - 4x
g = -4/2
g = -2
2fy = 14y
f = 14/2
f = 7
c = 28
r = √(-2)²+7²-28
r = √4+49-28
r = √25
r = 5
comparing centre (a,b) to (-g, -f)
a = -g = -(-2)
a = 2
b = -f = -7
b = -7
substituting the centre (a, b) and the radius r into the equation
(x-a)²+(y-b)² = r²
The required equation will be (x-2)²+(y+7)² = 5²
(x-2)²+(y+7)² = 25
A builder wants to install a triangular window with the angles shown in the illustration. What angles will he have to cut to make the window fit? (Hint: The sum of the angles in a triangle equals 180°.)
x° =
°
(x + 30)° =
°
Answer:
X=40°
Step-by-step explanation:
40+30=70
40+30=70
70+70=140
140+40=180
The angles will builder has to cut to make the window fit is 40°, 70° and 70°.
Given that, the measures of the three angles of a triangle are x°, (x+30)° and (x+30)°.
What is the angle sum property of a triangle?The sum of all internal angles of a triangle is always equal to 180°. This is called the angle sum property of a triangle.
Now, x°+(x+30)°+(x+30)°= 180°
⇒3x+60°=180°
⇒3x=180°-60°
⇒3x=120°
⇒x=40°
So, x°=40°, (x+30)°= 70°
Therefore, the angles will builder has to cut to make the window fit is 40°, 70° and 70°.
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I need help solving please do not report this question I need the answer I will give 30 points a heart 5.0 stars and brainlest
Answer:
280 m³
Step-by-step explanation:
Volume of a Triangular Prism
V = 1/2BHLSolving
V = 1/2 x 7 x 8 x 10V = 7 x 10 x 4V = 280 m³Answer:
280
Step-by-step explanation:
So first lets establish the formula for a Triangular Prism :
1/2 * b * h * l
now lets use the mesurements given to substitute into the formula :
1/2 * 7 * 8 * 10
now lets solve it by first multiplying all the whole numbers together!
7 * 8 = 56
56 * 10 = 560
Finally lets multiply 560 by 1/2 or in other words, divide 560 by 2 :
560/2 = 280
Pythagorean Theorem
solve for the missing side . Round to the nearest 10th
I NEED HELP ASAP I WILL MARK BRAINLIEST IF ANSWERED CORRECTLY NO LINKS IF YOU DO NOT KNOW THE ANSWER DONT COMMENT ON THIS THREAD
Answer:
\sqrt{45}
Step-by-step explanation:
A^2+B^2=C^2
36+?=81
81-36=45
[tex]\sqrt{45}[/tex]=
Answer:
6.7 mmStep-by-step explanation:
Pythagorean Theorem
a² + b² = c²Here, we are missing one of the altitudes, so we can arrange the equation to form :
a² = c² - b²Solving
UW² = VW² - UV²UW² = (9)² - (6)²UW² = 81 - 36UW² = 45UW = √45UW = 6.7 mmGiven two side lengths of a triangle (4cm and 17cm), find a range of possible lengths for the third side.
Check the picture below.
so notice the top part of the picture
if we move the "4" and "17" sides close to each other, till one is on top of the other, the side in red, can never be less than 17 - 4, or 13, it cannot be 13 either, because then both lines will be on on top and we wouldn't have a triangle just a flat-line, so whatever the red line is, it must be more than 13.
notice the bottom part of the picture
if we separate the "4" and "17" sides, and lay them flat, the most the red side can be is 4 + 17 or 21, it can never be greater than 21, it can't be 21 either because if we lay "4" and "17" flat, we lose the triangle and all we'd have is a flat-line. So whatever the red side is, must be less than 21.
Can someone please show step by step to this problem? Find the area of the following figure.
Answer:
20 x 12=240
Step-by-step explanation:
The rectangles are similar Find X.
The solution is Option A.
The value of x is given from proportion and x = 10 1/2 or x = 10.5
What is Proportion?
The proportion formula is used to depict if two ratios or fractions are equal. The proportion formula can be given as a: b::c : d = a/b = c/d where a and d are the extreme terms and b and c are the mean terms.
The proportional equation is given as y ∝ x
And , y = kx where k is the proportionality constant
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Given data ,
Let the first rectangle be A
The dimensions of the rectangle A = 4 by 3
Let the second rectangle be B
The dimensions of the rectangle B = 14 by x
The two rectangles A and B are similar , so
Now , the proportion equation will be
3 / 4 = x / 14
Multiply by 14 on both sides of the equation , we get
x = ( 14 × 3 ) / 4
x = 42/4
x = 10.5
The value of x = 10 1/2
Therefore , the value of x is 10 1/2
Hence , the value of x of the rectangle is 10 1/2
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what is 2/3 - 1/2 = in simplest form
Answer:
1/3
Step-by-step explanation:
Answer:
1/6
Step-by-step explanation:
2/3-1/2 = find the denominator 3×2=6
4/6-3/6 = rewrite with a denominator of 6
(4-3)/6 = 1/6 subtract numerators
The speed of the jet Alyssa flew on from Boston to London was 480 mph. On her return flight, the speed of the jet was 600 mph with a tail wind. What was Alyssa's average speed on the round trip? Round your answer to one decimal place.
Answer:
540
Step-by-step explanation:
Answer:
533.3mph
Step-by-step explanation:
A certain flight arrives on time 80
percent of the time. Suppose 131
flights are randomly selected. Use the normal approximation to the binomial to approximate the probability that
(a) exactly 103
flights are on time.
(b) at least 103
flights are on time.
(c) fewer than 98
flights are on time.
(d) between 98
and 117
,
inclusive are on time.
Answer:
B at least 103 flights are on time. :)
Step-by-step explanation:
First we need to turn the percentage into a decimal :)
131 x 0.80 = 104.8
The answer should be B at least 103 flights are on time.
Have an amazing day!!
Please rate and mark brainliest!!
please help me find the total area of the composite figure below (geometry)
Answer:
lw + [tex]\frac{1}{2}[/tex] × π × [tex](\frac{l}{2} )^{2}[/tex] ⇒ Answer D is correct
Step-by-step explanation:
First, let us find the area of the semi-circle.
Area = [tex]\frac{1}{2}[/tex] × π × r²
Given that,
diameter of the semi-circle is ⇒ l
∴ radius ⇒ l / 2
Let us find it now.
Area = [tex]\frac{1}{2}[/tex] × π × r²
Area = [tex]\frac{1}{2}[/tex] × π × [tex](\frac{l}{2} )^{2}[/tex]
Secondly, let us find the area of the rectangle.
Area = length × width
Given that,
length ⇒ l
width ⇒ w
Let us find it now.
Area = length × width
Area = l ×w
Area = lw
And now let us find the total area.
Total area = Area of the rectangle + Area of the semi - circle
Tota area = lw + [tex]\frac{1}{2}[/tex] × π × [tex](\frac{l}{2} )^{2}[/tex]
What is the surface area?
5 mm
5 mm
5 mm
square millimeter
I WILL GIVE 20 POINTS TO THOSE WHO ANSWER THIS QUESTION RIGHT NOOOO SCAMS AND EXPLAIN WHY THAT IS THE ANSWER
Recall: SOHCAHTOA
Sin M = Opp/Hyp
Reference angle = M
Opp = 3√/21
Hyp = 15
Sin M = (3-√21)/15
Sin M = √21/5
Cos M = Adj/Hyp
Reference angle = M
Adj = 6
Hyp = 15
Cos M = 6/15
Cos M = 2/5
Tan M = Opp/Adj
Reference angle = M
Opp = 3√21
Adj = 6
Tan M = (3√21)/6
Tan M = √21/2
:Therefore: Sin M = √21/5
Cos M = 2/5
Tan M = √21/2
Answer:
[tex]\sf \sin M=\boxed{\dfrac{\sqrt{21}}{5}}[/tex]
Step-by-step explanation:
Trigonometric ratios
[tex]\sf \sin(\theta)=\dfrac{O}{H}\quad\cos(\theta)=\dfrac{A}{H}\quad\tan(\theta)=\dfrac{O}{A}[/tex]
where:
[tex]\theta[/tex] is the angleO is the side opposite the angleA is the side adjacent the angleH is the hypotenuseGiven:
[tex]\theta[/tex] = MO = 3√21A = 6H = 15[tex]\sf \sin M=\dfrac{3\sqrt{21}}{15}\quad\cos M=\dfrac{6}{15}\quad\tan M=\dfrac{3\sqrt{21}}{6}[/tex]
[tex]\implies \sf \sin M=\dfrac{3\sqrt{21}}{15}=\dfrac{\sqrt{21}}{5}[/tex]
What is the equation of this line in slope-intercept form?
y= 3x-1
y = 3x+1
y= - 1/3x-1
y = - 3x-1
Answer:
Step-by-step explanation:
y=-3x-1
Please solve the following thank you so much math experts
Answer:
4) 97%
6) 13%
Step-by-step explanation:
4) We are given that 3% for 250 million are diesel.
100% - 3% = 97% NON DIESEL cars
97% of 250 million is 242.45 million.
______________________________
6) We know that 17% (buses) and 70% (cars) are part of that total.
We also know that the total needs to be 100%
So, we simply do this:
100 - (17+70) = 13 (%)
So, 13% of the survey participants had bikes
Make a stem-and-leaf plot of the data values 14, 22, 9, 13, 30, 8, 25, and 29.
Stem Leaf
89
1 3
22
0
Key: 1 | 3 =
Previous
H
9
2
3
SO
Answer:
Step-by-step explanation:
BRAINLIEST + STARSS!!
In this diagram, x || y || z and m<9 = 93°.
What is the measure of each angle?
Drag and drop the correct measurements into the boxes to
match each angle.
m<2
m<4
m<7
89° 91 ° 93 ° 87°