I'm assuming the integral is
[tex]\displaystyle \int \frac{\mathrm dx}{x+\sqrt{x^2-1}}[/tex]
Rationalize the denominator:
[tex]\dfrac1{x+\sqrt{x^2-1}} \times \dfrac{x-\sqrt{x^2-1}}{x-\sqrt{x^2-1}} = \dfrac{x-\sqrt{x^2-1}}{x^2-\left(\sqrt{x^2-1}\right)^2} = x-\sqrt{x^2-1}[/tex]
Then the integral is
[tex]\displaystyle \int\left(x-\sqrt{x^2-1}\right)\,\mathrm dx = \dfrac12x^2 - \int\sqrt{x^2-1}\,\mathrm dx[/tex]
For the remaining integral, substitute x = sec(t ) and dx = sec(t ) tan(t ) dt. Then over an appropriate domain, we have
[tex]\displaystyle\int\sqrt{x^2-1}\,\mathrm dx = \int\sec(t)\tan(t)\sqrt{\sec^2(t)-1}\,\mathrm dt = \int\sec(t)\tan^2(t)\,\mathrm dt[/tex]
Integrate by parts, taking
u = tan(t ) ==> du = sec²(t ) dt
dv = sec(t ) tan(t ) dt ==> v = sec(t )
Then
[tex]\displaystyle\int\sec(t)\tan^2(t)\,\mathrm dt = \sec(t)\tan(t) - \int\sec^3(t)\,\mathrm dt[/tex]
Now for *this* remaining integral, integrate by parts again, taking
u = sec(t ) ==> du = sec(t ) tan(t ) dt
dv = sec²(t ) dt ==> v = tan(t )
so that
[tex]\displaystyle\int\sec^3(t)\,\mathrm dt = \sec(t)\tan(t) - \int\sec(t)\tan^2(t)\,\mathrm dt \\\\ \int\sec^3(t)\,\mathrm dt = \sec(t)\tan(t) - \int\sec(t)(\sec^2(t)-1)\,\mathrm dt \\\\ \int\sec^3(t)\,\mathrm dt = \sec(t)\tan(t) - \int\sec^3(t)\,\mathrm dt + \int\sec(t)\,\mathrm dt \\\\ \int\sec^3(t)\,\mathrm dt = \frac12\sec(t)\tan(t)+\frac12\int\sec(t)\,\mathrm dt \\\\ \int\sec^3(t)\,\mathrm dt = \frac12\sec(t)\tan(t) + \frac12\ln\left|\sec(t)+\tan(t)\right| + C[/tex]
To summarize, if I denotes the original integral, we have
[tex]\displaystyle I = \frac12x^2 - \int\sqrt{x^2-1}\,\mathrm dx \\\\ I = \frac12x^2 - \int\sec(t)\tan^2(t)\,\mathrm dt \\\\ I = \frac12x^2 - \sec(t)\tan(t) + \int\sec^3(t)\,\mathrm dt \\\\ I = \frac12x^2 - \sec(t)\tan(t) + \frac12\sec(t)\tan(t) + \frac12\ln\left|\sec(t)+\tan(t)\right| + C \\\\ I = \frac12x^2 - \frac12\sec(t)\tan(t) + \frac12\ln\left|\sec(t)+\tan(t)\right| + C[/tex]
Putting everything back in terms of x, we have
sec(t ) = x
tan(t ) = √(x ² - 1)
so that
[tex]\displaystyle I = \boxed{\frac12x^2 - \frac12x\sqrt{x^2-1}+\frac12\ln\left|x+\sqrt{x^2-1}\right|+C}[/tex]
find k so that x^2+2x+k is a factor of 2x^4+x^3-14x^2+5x+6. also find all the zeroes of the two polynomial
Compute the quotient and remainder,
[tex]\dfrac{2x^4+x^3-14x^2+5x+6}{x^2+2x+k} \\\\ = 2x^2 - 3x - (8+2k) + \dfrac{(21+7k)x+(6+8k+2k^2)}{x^2+2x+k}[/tex]
The remainder upon dividing [tex]2x^4+x^3-14x^2+5x+6[/tex] by [tex]x^2+2x+k[/tex] should leave no remainder, which means
[tex]21+7k = 0 \implies 21 = -7k \implies k=-3[/tex]
and
[tex]6+8k+2k^2 = 0 \implies 2(k+3)(k+1)=0 \implies k=-3\text{ or }k=-1[/tex]
Only k = -3 makes both remainder terms vanish.
Then the previous result reduces to
[tex]\dfrac{2x^4+x^3-14x^2+5x+6}{x^2+2x-3} = 2x^2 - 3x - 2[/tex]
so that
[tex]2x^4+x^3-14x^2+5x+6 = (x^2+2x-3) (2x^2 - 3x - 2) \\\\ 2x^4+x^3-14x^2+5x+6 = (x+3)(x-1)(2x + 1)(x-2)[/tex]
and so the zeroes of the quartic polynomial are x = -3, x = 1, x = -1/2, and x = 2.
When -2(x − 4) + 1 = 7 is solved, the result is:
Answer:
1
Step-by-step explanation:
-2(x-4)+1=7
First Distribute the -2.
-2x+8+1=7
Subtract the 8 and 1 from the whole equation.
-2x=-2
Divide both sides of the equation by -2.
x=1
I hope this helps!
Answer:
x = 1
Step-by-step explanation:
-2(x − 4) + 1 = 7
-2x + 8 + 1 = 7
-2x + 9 = 7
-2x = -2
x = 1
What is the area? please help
Answer:
70.875 [tex]km^{2}[/tex]
Step-by-step explanation:
Using the equation of area of trapezium
[tex]\frac{1}{2} * (a+b)h[/tex]
Put the value into the equation
[tex]\frac{1}{2}(4.8+5.7)*13.5[/tex]
Simplify the equation and you will get:
[tex]2.4*13.5=70.875 km^{2}[/tex]
So, the answer is 70.875km^2
50 points and brainly
Trinas you company charges $0.86 per pound to ship toys. Part A: Write an equation to determine the totals cost, c, to ship p pounds of toys. Use your equation to determine the cost of shipping 2 pounds of toys. Path B: if the company reduces the cost to ship toys by $0.05 per pound, write an equation to determine the total cost, c, to ship p pounds of toys with the reduced cost.
Answer:
Part A:
c = .86*p
c =1.72 for 2 lbs
Part B:
c =.81p
Step-by-step explanation:
Part A:
Total cost = cost per pound * number of pounds
c = .86*p
Let p = 2
c = .86*2
c =1.72
Part B:
Total cost = cost per pound * number of pounds
c = .(.86-.05)*p
c =.81p
Which answer shows 0.00897 written in scientific notation?
From what I heard it's not D
Answer:
B
Step-by-step explanation:
The decimal number 0.00897 written in scientific notation is 8.97 ×10^3 and it has significant figures
Answer:
8.97 * 10⁻³
Step-by-step explanation:
0.00897 = [tex]\frac{8.97}{1000}=8.97 * 10^{-3}[/tex]
Place the decimal point after the first nonzero number.
Write the power of 10,
(i) If we move the decimal to the right, the power of 10 decrease by number of moves.
(ii) If we move the decimal to the left, the power of 10 increase by the number of moves
Evaluate the expression 34 ÷ (14 − 5) × 2.
I'm pretty confused can someone explain it to me? I will give branliest
Answer:
[tex]7\frac{5}{9}[/tex]
Step-by-step explanation:
Hi there!
We are given this expression:
34÷(14-5)×2
And we want to evaluate it
We'll be using the order of operations
Depending on the country you're in, the acronym can be different, but in the US, the typical acronym used for the order of operations is PEMDAS, or Parentheses, Exponents, Multiplication, Division, Addition, Subtraction
The order of operations states that we should start with any expressions in parentheses
In this case, it's (14-5), which is equal to 9
The expression now becomes
34÷9×2
There are no exponents, so let's skip that
Now this is where it gets tricky:
Based on the acronym alone, we should be doing 9*2 next, however, we should always work left to right, even if in a case like this, we'll be doing division instead of multiplication
Divide 34 by 9
[tex]\frac{34}{9}[/tex]×2
Now multiply [tex]\frac{34}{9}[/tex] by 2
[tex]\frac{68}{9}[/tex]
(as a mixed fraction, it's 7[tex]\frac{5}{9}[/tex])
Hope this helps!
Name the property shown by the statement 2a = a2
Distributive property
screenshot if you need more words. seems to be a well illustrated and compact explanation right there
The area of a square piece of land is 784metersquared .What is the perimeter
Answer:
112 m
Step-by-step explanation:
The area (A) of a square is
A = s² ( s is the length of side )
Here A = 784 , then
s² = 784 ( take the square root of both sides )
s = [tex]\sqrt{784}[/tex] = 28
Then
perimeter = 4s = 4 × 28 = 112 m
Answer:
112m
Step-by-step explanation:
the area of the square is 784m^2 so the square root of 784 would give us the length of the sides(since it's a square the length of each sides are equal). √784 would give 28m. 28 m is the length of each sides..so now the perimeter would be the addition of each sides which in case would be 4 times 28 because it's a square. 4 times 28 gives 112m.
(4x-3)(x-5)-2x(2x-11)
Answer:
3(13x-5)
Step-by-step explanation:
first step open your bracket,u have
=4x²+20x-3x-15-4x²+22x then collect like terms
=4x²-4x²+20x-3x+22x-15
=0x²+39x-15
=39x-15
further simplifying u have:
=3(13x-5)
Find the value of each measure.
Answer:
39
Step-by-step explanation:
4x+7=9x-33
4x-9x=-33-7
-5x=-40
x=8
4×8+7=39
Given the center and radius, find equation of the circle
Answer:
[tex]{ \sf{ {(x - 2)}^{2} + {(y + 8)}^{2} = 4}}[/tex]
Answer:
Center is (2, -8) and radius =2
h = 2 and k = -8
Filling in this equation:
(x -h)² + (y -k)² = radius² we get:
(x - 2)^2 + (y--8)^2 = 2^2 which equals
(x-2)^2 + (y +8)^2 = 4
Source: http://www.1728.org/circleqn.htm
Step-by-step explanation:
Help fast pleaseeeeeeee
Step-by-step explanation:
-2. = [tex] \frac{1}{49} [/tex]-1 = [tex] \frac{1}{7} [/tex]0 = 11. = 72. = 496. How much larger is 2 x 104 than 8 x 102?
A. 15
B. 20
C. 25
D. 30
Answer:
20
Step-by-step explanation:
it is 20
Please help i do not understand
Answer:
OPTION A
I CHOSE OPTION A AS THE ANSWER BECAUSE OPTI0N D CANNOT BE TRUE ACCORDING TO THE QUESTION AND OPTION B AND C ARENT TRUE BECAUSE IT WAS MENTIONED THAT AT 3 HOURS THE COST WOULD BE 140 $ BUT IN OPTION B IT SHOWS THAT THEY ALREADY ACHIEVED 200$ BEFORE IT EVEN BECAME 3 HOURS AND IT ISNT OPTION C BECAUSE IT CROSSED 150$ WHILE IT WAS 3 HOURS SO MOST LIKELY THE ANSWER IS TO BE OPTION A.
HOPE THIS HELPS YOU....
Need help ASAPPPPPPP
Answer:
A) 5
Step-by-step explanation:
51+48+15x+6=180
105+15x=180
15x=75
x=5
The side lengths of a 45-45-90 triangle are in the ratio 1:1:√2. What is sin 45°?
Answer:
[tex]\frac{\sqrt{2} }{2}[/tex]
Step-by-step explanation:
To find the value of sin 45° we divide the opposite side length with hypotenuse
[tex]\frac{1}{\sqrt{2} }[/tex] we then multiply this by [tex]\sqrt{2}[/tex] because denominator cannot be irrational
[tex]\frac{1}{\sqrt{2} }[/tex] x [tex]\sqrt{2}[/tex] = [tex]\frac{\sqrt{2} }{2}[/tex] gg
Anyone Answer Please
Answer:
shift of 6 units left
Step-by-step explanation:
Given f(x) then f(x + a) is a horizontal translation of f(x)
• If a > 0 then a shift left of a units
• If a < 0 then a shift right of a units
f(x) = 3 | x + 6 | + 9
Represents a horizontal shift of the parent function 6 units left
find the shaded area...
Answer:
148°
Step-by-step explanation:
The central angle is twice the angle at the circumference, subtended on the same arc.
shaded angle = 2 × 74° = 148°
Two friends are writing practice problems to study for a trigonometry test. Sam writes the following problem for his friend Anna to solve:
In right triangle ABC, the measure of angle C is 90 degrees, and the length of side c is 8 inches.
Solve the triangle.
Anna tells Sam that the triangle cannot be solved. Sam says that she is wrong.
Who is right? Explain your thinking
Answer:
Anna is right in her meaning concerning on triangle solvability.
Step-by-step explanation:
The side [tex]c[/tex] represents the hypotenuse of a right triangle as [tex]C = 90^{\circ}[/tex] and is opposite to that angle. There are two ways to solve this triangle trigonometrically:
i) Law of Sine
[tex]\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}[/tex] (1)
ii) Law of Cosine
[tex]c^{2} = a^{2} + b^{2} - 2\cdot a\cdot b \cdot \cos C[/tex] (2)
The Pythagorean Theorem is a particular case of the Law of Cosine for [tex]C = 90^{\circ}[/tex]
The triangle cannot be solved as there is an input missing, either another side or another angle. If [tex]C = 90^{\circ}[/tex], then (2) is reduced into this form:
[tex]c^{2} = a^{2}+b^{2}[/tex] (2b)
In this case we need to know the measure of either [tex]a[/tex] or [tex]b[/tex] to determine its counterpart and the values of the missing angles by (1). In nutshell, Anna is right.
The times to pop a 3.4-ounce bag of microwave popcorn without burning it are Normally distributed with a mean
time of 140 seconds and a standard deviation of 20 seconds. A random sample of four bags is selected and the
mean time to pop the bags is recorded. Which of the following describes the sampling distribution of all possible
samples of size four?
This question is solved using the central limit theorem, giving an answer of:
Fourth option, approximately normal with mean of 140 seconds and standard deviation of 10 seconds.
Central Limit Theorem
The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
Mean of 140, standard deviation of 20, sample of 4:
By the Central Limit Theorem, the distribution is approximately normal.
Mean is the same, of 140.
[tex]n = 4, \sigma = 20[/tex], thus:
[tex]s = \frac{\sigma}{\sqrt{n}} = \frac{20}{\sqrt{4}} = 10[/tex]
Thus, the correct answer is:
Fourth option, approximately normal with mean of 140 seconds and standard deviation of 10 seconds.
For another example of the Central Limit Theorem, you can check https://brainly.com/question/15519207
Given m
11
n, find the value of x and y.
41°
וז.
20
n
yº
So I keep getting the wrong answer could someone help me pls
Answer:
x = 41°, y = 139°
Step-by-step explanation:
The given parameters are;
Line m is parallel to line n and lines m and n have a common transversal
The corresponding angles formed by the common transversal to the two parallel lines are 41° on line m and x° on line n
Therefore, x° = 41° by corresponding angles formed between on two parallel lines by a common transversal are equal
x° and y° are linear pair angles and they are, supplementary
∴ x° + y° = 180°
∴ x° + y° = 41° + y° = 180°
y° = 180° - 41° = 139°
y° = 139°.
Solve the quadratic equation by completing the square.
Answer:
[tex] {k}^{2} + 2k + \frac{15}{2} = 0 \\ {k}^{2} + 2k + 1 = - \frac {15}{2} + 1 \\ {k + 1}^{2} = - \frac{13}{2} \\ k + 1 = - \sqrt{ \frac{13}{2} } \\ k = 1 + or - - \sqrt{ \frac{13}{2} } [/tex]
given a sphere with a radius of m. what is the locus of the midpoints of the radii of the sphere?
Answer:
The locus of the midpoints of the radii of the sphere is a sphere half the radii M of the given sphere and having the same center as of the given sphere.
Step-by-step explanation:
please answer and no links please ....
Step-by-step explanation:
ℎ = -4, -3×(-4)+2 = 12+2 = 14
ℎ = -2, -3×(-2)+2 = 6+2 = 8
ℎ = 0, -3×0+2 = -3+2 = 2
ℎ = 3, -3×3+2 = -9+2 = -7
ℎ = 5, -3×5+2 = -15+2 = -13
factor and solve the problem in the photo ……. pleaseeee helppppp i havent done algebra in 2 years
Answer:
Not factarable, all terms must be in x. or y
Step-by-step explanation:
PLZZZ HELPPP, IF NOT 100% SURE PLZ DONT ANSWER…BRAINLIEST TO FIRST AND CORRECT ANSWER, THX TO SECOND AND CORRECT ANSWER
Colin is painting figurines. He spends 20 minutes painting each figurine. After painting for 60 minutes, he still has 9 more figurines left to paint.
Answer:
9 times 20 is equally to 180 minutes
Answer:
f=-1/20t+12
Step-by-step explanation:
khan
Which constants can be multiplied by the equations so one variable will be eliminated when the systems are added together?
5x + 13y = 232
12x + 7y = 218
The first equation can be multiplied by –13 and the second equation by 7 to eliminate y.
The first equation can be multiplied by 7 and the second equation by 13 to eliminate y.
The first equation can be multiplied by –12 and the second equation by 5 to eliminate x.
The first equation can be multiplied by 5 and the second equation by 12 to eliminate x.
The volume of a Rubik’s cube is 512/216 cubic inches. Find the length of a side of the cube.
Answer:
4/3 inches, or 1.33333 inches
Step-by-step explanation:
In terms of the volume of a cube, its side length is the cube root of it. Therefore, for a cube with a volume of 512/216 cubic inches, its side length is equal to ∛(512/216) = ∛(512)/∛(216) = 8/6 = 4/3 inches*
* we know that 8/6 = 4/3 because we can multiply both the numerator and denominator by 1/2 to get this result. If we multiply/divide the numerator or denominator by the same values, the fraction is still the same
Find the measure of <3. 90 50 40 130
=============================================================
Explanation:
This figure is a kite since we have two pairs of adjacent sides that are congruent, but not all sides are the same length.
One property of kites is that the diagonals are always perpendicular (this applies to rhombuses as well). This means angle 3 is 90 degrees.
The measure of ∠3 is 90°
Because in a kite the diagonals bisects at 90°
Therefore ∠3 is 90°
Answered by Gauthmath must click thanks and mark brainliest
