The value of the trigonometric functions cos 2θ and cos 0.5θ for the given angle are - 7 / 25 and - 2√5 / 5, respectively.
How to determine the exact values of trigonometric expressions
In this problem we must determine the cosine of the double angle and half angle based on the exact values of the cosine of an angle and the quadrant location of the angle.
Cosine is a function between - 1 and 1 and whose period is 2π (360°), positive when angle is either on first quadrant (0 < θ < 0.5π) or on fourth quadrant (1.5π < θ < 2π). In addition, sine function is positive on first quadrant and negative on fourth quadrant.
The required trigonometric formulas in terms of trigonometric function cos θ are described below:
sin θ = √(1 - cos² θ)
cos 2θ = cos² θ - sin² θ
cos 0.5θ = ± √ [(1 + cos θ) / 2]
First, determine the exact value of sine:
sin θ = - √[1 - (3 / 5)²]
sin θ = 4 / 5
Second, determine the exact value of cosine of the double angle:
cos 2θ = (3 / 5)² - (4 / 5)²
cos 2θ = - 7 / 25
Third, determine the exact value of cosine of the half angle:
cos 0.5θ = - √[(1 + 3 / 5) / 2]
cos 0.5θ = - 2√5 / 5
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A hyperbola centered at (5, 0) has a focus at (5, 10) and vertex at (5, 6). Which is the equation of the hyperbola in standard form?
The equation of the hyperbola in standard form is y²/8² - (x-5)²/6² = 1
Given that the center of the hyperbola is (5,0) and the vertex is (5,6), we can see that the center is on the x-axis, so the y-coordinate of the center, k, is 0. The x-coordinate of the center, h, is 5.
We know that equation of Hyperbola is
(y-k)²/a² - (x-h)²/b² = 1
Where (h,k) is the center and given (h, k) = (5, 0) and b is the transverse axis and a is the conjugate axis.
Now b is the y-coordinate of vertex (5,6) ⇒b=±6
and c is the y-coordinate of focus (5,10)c=±10
And also c² = a²+b²
(±10)²=a²+(±6)²
⇒a²=100−36=64
Hence a=±8
Thus, the equation of the hyperbola is y²/8² - (x-5)²/6² = 1
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exponent rules: (6m⁸n⁹) (4mn⁷) = ?
please help
The value of the given equation is, (6m⁸n⁹) (4mn⁷) =[tex]$$=24 m^9 n^{16}$$[/tex]
What is exponent rules?(am)n = am*n is the exponent power rule. Exponent times power is how you raise a number with an exponent to a power. The x-n = 1/xn Negative Exponent Rule. A negative exponent can become positive by inverting the base. According to the first law, adding the exponents will multiply two exponential functions with the same base. According to the second law, we must subtract the exponents when dividing two exponential functions with the same base. The third law indicates that we must multiply the exponents in order to increase a power to a new power.Simplify[tex]$\left(6 m^8 n^9\right)\left(4 m n^7\right): \quad 24 m^9 n^{16}$[/tex]
steps
[tex]$$\left(6 m^8 n^9\right)\left(4 m n^7\right)$$[/tex]
Apply exponent rule: [tex]$a^b \cdot a^c=a^{b+c}=0^{\prime}=$[/tex]
[tex]& m^8 m=m^{8+1} \\[/tex]
[tex]& =6 n^9 \cdot 4 m^{8+1} n^7[/tex]
\end{aligned}
Add the numbers: [tex]$8+1=9$[/tex]
[tex]=6 n^9 \cdot 4 m^9 n^7[/tex]
Apply exponent rule: [tex]$a^b \cdot a^c=a^{b+c}$[/tex]
[tex]& n^9 n^7=n^{9+7} \\[/tex]
[tex]& =6 \cdot 4 m^9 n^{9+7}[/tex]
Add the numbers: [tex]$9+7=16$[/tex]
[tex]$$=6 \cdot 4 m^9 n^{16}$$[/tex]
Multiply the numbers: 6 - [tex]$4=24$[/tex]
[tex]$$=24 m^9 n^{16}$$[/tex]
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The following table shows the total number of quizzes taken in a high school.
4 quizzes are taken per week.
What is a linear equation?
A linear equation is an algebraic equation of the form y=mx+b. involving only a constant and a first-order (linear) term, where m is the slope and b is the y-intercept.
A linear equation is given by: y = mx + b;
where y, x are variables, m is the rate of change and b is the y intercept.
Let y represent the total number of quizzes after x weeks.
From the table we can see that when y = 12, x = 3 and when y = 20, x = 5. Hence:
Number of quizzes per week = 12/3 = 20/5 = 4
Hence, 4 quizzes are taken per week.
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please answer.........................
The proof of <BED = <BAC is shown below.
What is Similarity?If two triangles have an equal number of corresponding sides and an equal number of corresponding angles, then they are comparable. Similar figures are described as items with the same shape but varying sizes, such as two or more figures.
Given:
AE ⊥ BC
and, CD ⊥ AB
As, AE is perpendicular to BC then BE = EC = 1/2 BC
and, BD = AD = 1/2 AB
Now, In ΔBED and ΔBCA
<EBD = <ABC {common}
BD/AB = BE/ BC = 1/2
By ASA similarity Criteria ΔBED ~ ΔBCA
So, <BED = <BAC
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Which one of the following is used in predictive analytics? a. Data visualization b. Linear regression c. Data dashboard d. Optimization model
Linear regression is a statistical tool used in predictive analytics to identify the relationship between a dependent variable and one or more independent variables.
Linear regression is a statistical technique used in predictive analytics to identify the relationship between a dependent variable and one or more independent variables. It is used to predict future outcomes by analyzing data from the past. It works by fitting a linear equation to the data, which is then used to estimate the value of the dependent variable for any given combination of values of the independent variables. The linear equation is constructed by finding the line of best fit through the data points. Linear regression can be used to identify trends and patterns in the data, and to make predictions about future outcomes. It is a powerful tool for predicting the future, and can be used to make informed decisions about how to best use resources.
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use the ALEKS calculator to write 31/24 as a decimal rounded to the nearest hundredth.
The decimal form of the value, 31/24 is 1.29166... The nearest hundredth form of this expression will be 1.29.
What is the nearest hundredth?To calculate the nearest hundredth you have to examine the second number that comes after the decimal point. To get the nearest hundredth, leave the hundredth value as it is if the value after it is not up to five. If the value after the hundredth value is more than 5, then we have to add 1 to the figure that constitutes the hundredth value.
In the question given above, 9 is the hundredth value. The value after 9 is 1 and since it is less than 5, it will not be changed. Therefore, the figure when rounded off to the nearest hundredth form is 1.29.
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someone please help me I need to return this tomorrow ;(
Finnd the missing dimension. Show how you found the answer.
Answer: 15
Step-by-step explanation:
Area = bh
Area = 120
Height = 8
Since you have the area and the height you need to divide:
120/8
=
15
__________________________________________________________
You can also check your answer by doing bh
15 x 8
=
120
2) what is the domain of f(x) = x³ + 6x² + 5
Answer:
[tex]-\infty \: < x < \infty[/tex]
or, in interval notation as
[tex]\left(-\infty \:,\:\infty \:\right)[/tex]
Step-by-step explanation:
The domain of a function [tex]f(x)[/tex] is the set of all values of [tex]x[/tex] for which [tex]f(x)[/tex] is real and defined
The given function is f(x) = x³ + 6x² + 5
There are no undefined points. So the domain is the set of all real numbers that can be expresses as
[tex]-\infty \: < x < \infty[/tex]
or, in interval notation
[tex]\left(-\infty \:,\:\infty \:\right)[/tex]
PLEASE HELP ASAP!!!!!!!!
You must be more than 3.5 feet tall to ride this ride. Write an inequality to represent
this. Use x as your variable.
Inequality compares two variables, to show if one is less than, equal to or greater than the other variable being compared.
How to calculate inequality?inequality becomes h>3.5When solving an inequality: • you can add the same quantity to each side • you can subtract the same quantity from each side • you can multiply or divide each side by the same positive quantity If you multiply or divide each side by a negative quantity, the inequality symbol must be reversed.Subtract the same number from both sides. Multiply both sides by the same positive number. Divide both sides by the same positive number. Multiply both sides by the same negative number and reverse the sign.Rule 1. Adding or subtracting the same quantity from both sides of an inequality leaves the inequality symbol unchanged. Rule 2. Multiplying or dividing both sides by a positive number leaves the inequality symbol unchanged.To learn more about inequality refers to:
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g(x) = 6 - x; then g(-4) = _____
Answer: 10
scince x is equal to -4 the equation becomes g(-4)=6- -4
What is quadratic equation examples with answers?
A quadratic equation is an equation of degree 2 and has the standard form: ax² + bx + c = 0, with a ≠ 0. Example: 2x² - 3x - 2 = 0, the solution is {-1/2, 2}.
A quadratic equation is an equation of degree 2. The standard form of a quadratic equation is:
ax² + bx + c = 0, with a ≠ 0
The quadratic equations can be solved using:
- factorization
- completing the square
- abc formula
Example 1:
2x² - 3x - 2 = 0
Using factorization:
2x² - 3x - 2 = (2x +1) (x-2)
Hence,
(2x +1) (x-2) = 0
Solution:
x = -1/2 and x = 2
x = {-1/2, 2}
Example 2:
x² - 6x + 4 = 0
By completing the square:
x² - 6x + 5 = (x - 3)² - 4
Hence,
x² - 6x + 5 = 0
(x - 3)² - 4 = 0
(x - 3)² = 4
x - 3 = ± 2
The solutions are
x = 3 - 2 = 1 or
x = 3 + 2 = 5
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PLEASE HELP ME, THANK YOU!!!!!
The standard form of the given polynomial is x⁶ + 53x⁵ + 2x⁴ - 2 which has a highest degree of 6 while the leading coefficient is 1. This polynomial is called a quadrinomial because it has 4 terms
What is a PolynomialA polynomial is an equation that only uses addition, subtraction, multiplication, and non-negative integer exponents of variables and consists of variables (also known as indeterminates) and coefficients. Algebra, calculus, and numerical analysis are just a few of the mathematics and scientific disciplines that use polynomials.
In the presented problem, we must determine the terms, variables, coefficients, and other properties of the polynomial f(x).
f(x) = 2x⁴ + 53x⁵ - 2 + x⁶
The standard form of this polynomial is x⁶ + 53x⁵ + 2x⁴ - 2
The degree of the polynomial is 6
The leading coefficient is 1 which is the coefficient of the degree of the polynomial
This is called a Quadrinomial because it has 4 terms
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The salary of 157 employees in an office for a month is Rs. 7,10582.What is the annual salary of 1 employee, assuming all are paid equally?
Show division step-by-step
Answer:
To find the annual salary of one employee, you need to divide the total monthly salary by the number of employees and then multiply by 12 to get the annual salary.
Here's the step-by-step process:
Divide the total monthly salary by the number of employees: 710582 / 157 = 4550.85
Multiply the result by 12 to get the annual salary: 4550.85 * 12 = Rs. 54610.20
So, the annual salary of one employee is Rs. 54610.20.
Step-by-step explanation:
Show that f is continuous on (−[infinity],[infinity])f(x)=1−x^2 if x≤1ln x if x>1
the centroid of the region bounded by the given curves is (x, y) = (15, 225).
To show that f is continuous on (−∞,∞), we need to show that the limit of f(x) as x approaches any point from the left and from the right is equal.
First, let's consider the limit of f(x) as x approaches 1 from the left.
As x approaches 1 from the left, f(x) = 1-x^2, so the limit of f(x) as x approaches 1 from the left is 1.
Now, let's consider the limit of f(x) as x approaches 1 from the right.
As x approaches 1 from the right, f(x) = ln x, so the limit of f(x) as x approaches 1 from the right is 0.
Since the limit of f(x) as x approaches 1 from the left is 1 and the limit of f(x) as x approaches 1 from the right is 0, we can conclude that f is continuous on (−∞,∞).
To find the centroid of the region bounded by the given curves, we need to calculate the area of the region and the area-weighted centroid of the region.
The area of the region is given by the integral of x3 from 0 to 10, which is equal to 1125/4.
The area-weighted centroid of the region is given by the integral of (x3*x)/(1125/4) from 0 to 10, which is equal to 15.
Therefore, the centroid of the region bounded by the given curves is (x, y) = (15, 225).
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PLEASE HELP ASAP!!!!!!!!
Write the equation of the line through the two points (1,-1) and (2,4).
Answer: First point 6x -2y and second point 3x 4y
Step-by-step explanation:
Work out the volume of each of this prisms:
The volumes of right prisms are listed below:
Case A: 150 cm³
Case B: 525 m³
Case C: 429 m³
How to determine the volume of a right prism
The volume of right prism (V), in cubic length units, is always equal to the product of the base area (A), in square length units, and height (h), in length units:
V = A · h
There are three right prisms, whose areas are combinations of right triangles and rectangles. The area formulas are shown below:
Rectangle
A = w · l
Right triangle
A = 0.5 · w · l
Where:
w - Width, in length units.l - Length, in length units.The complete procedure is listed below:
Determine the base area.Calculate the volume of prism.Case A
A = 0.5 · (4 cm) · (5 cm) + (3 cm) · (5 cm)
A = 25 cm²
V = (25 cm²) · (6 cm)
V = 150 cm³
Case B
A = (5 m) · (8 m) + (5 m) · (7 m)
A = 75 m²
V = (75 m²) · (7 m)
V = 525 m³
Case C
A = (8 m) · (4 m) + (3 m) · (9 m) + 0.5 · (5 m) · (5 m)
A = 71.5 m²
V = (71.5 m²) · (6 m)
V = 429 m³
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Find the value of x-
1250=1/2×50×(2x+3x)
The value of x in the expression is 10
How to find x?You should understand that an equation is a mathematical statement showing the equality of two things
The given equation is 1250=1/2×50×(2x+3x)
To find the value of x, we have to simplify the right hand side of the equation
This is = 1250 = 50(5x)/2
This will give us 2500 250x after cross multiplication
Then make x the subject of the relation by dividing by 250
This implies that x = 10
The expression leaves the value of x at 10
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the ratio of union to nonunion workers is 7 to 3. if there are 18 nonunion workers how, many union workers are there?
The number of union workers is 42
What is a ratio?
A ratio can be defined in mathematics as an ordered pair of numbers, like y and z, that is written in the form y / z such that the values of z is not equal to zero.
It also shows how many the times a number contains another number in a given proportion.
From the information given, we have that;
Ratio of union workers = 7Ratio of non-union workers =3Total number of non- union workers = 18Then,
If 3 = 18 workers
Then 7 = x workers
cross multiply
3x = 18(7)
Multiply the values
3x = 126
x = 42 workers
Hence, the number is 42
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what is the total dose required for a 140 lb patient if the amount required is 28 mg/kg bodyweight?
The total dose required for a 140 lb patient is 3,920 mg, as it is calculated by multiplying 28 mg/kg bodyweight.
1. Calculate the patient's bodyweight in kilograms by dividing their weight in pounds (140) by
2.2: 140 ÷ 2.2
= 63.63 kg
2. Multiply the patient's bodyweight in kilograms (63.63) by the required dose per kilogram (28):
63.63 x 28
= 1,780.84 mg
3. Round this number up to the nearest whole number: 1,780.84 rounded up = 3,920 mg
The total dose required for a 140 lb patient is calculated by multiplying the amount required per kilogram of bodyweight by the patient's bodyweight in kilograms. To calculate the patient's bodyweight in kilograms, you need to divide their weight in pounds (140) by 2.2. This gives you 63.63 kg. Then, multiply the patient's bodyweight in kilograms (63.63) by the required dose per kilogram (28): 63.63 x 28 = 1,780.84 mg. Finally, round this number up to the nearest whole number: 1,780.84 rounded up = 3,920 mg. Therefore, the total dose required for a 140 lb patient is 3,920 mg.
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I need help with this assignment and finding the measures and equations
Answer:
(4x-9)° +(6x+2)° +x° = 180°∠P = 59°∠Q = 104°∠R = 17°Step-by-step explanation:
Given ∆PQR with ∠P = (4x-9)°, ∠Q = (6x+2)°, and ∠R = x°, you want an equation to find x, and the measures of the angles.
EquationThe equation expresses the fact that the sum of angles in a triangle is 180°.
∠P +∠Q +∠R = 180°
(4x -9)° +(6x +2)° +x° = 180°
Solution11x -7 = 180 . . . . . . . . . . . . divide by °, collect terms
11x = 187 . . . . . . . . . add 7
x = 17 . . . . . . . . divide by 11
AnglesThe angle measures are found by substituting for x in the angle expressions:
∠P = (4·17 -9)° = 59°
∠Q = (6·17 +2)° = 104°
∠R = 17°
Imogen is organising a barbecue.
She needs bread rolls and burgers.
Bread rolls are sold in packs of 20.
Burgers are sold in packs of 12.
Imogen buys exactly the same number
of bread rolls as burgers.
What is the least number of each pack
that Imogen buys?
packs of bread rolls
............... packs of burgers
Imogen need to buy 3 packs of bread rolls for every 5 packs of Burgers packs that are sold.
What is Equation?Two or more expressions with an Equal sign is called as Equation.
X= Number of Bread Roll Packs
Y= Number of Burger Packs
Therefor we can say
20X=12Y
X/Y=12/20=3/5
Therefore Imogen need to buy 3 packs of bread rolls for every 5 packs of Burgers packs that are sold.
Hence, Imogen need to buy 3 packs of bread rolls for every 5 packs of Burgers packs that are sold.
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Eating at all you can eat buffet at your favorite restaurant cost $18.50, and increases its cost by 6% each year .
1. Write a function that correctly depicts the scenario.
2. How much will the buffet cost 10 years from now round your answer to the nearest cent
3. How much did the buffet cost 15 years ago ?round your answer to the nearest cent
The function that correctly depicts the scenario is represented as; f(x) = 18.50 (1.06)^x
The cost of the buffet in 10 years is; $33.13
The cost of the buffet 15 years ago is ; $7.71
What is a function?Function is a type of relation, or rule, that maps one input to specific single output.
Here we have the following parameters that can be used in our computation:
Value of buffet = $18.50
Rate of increment = 6%
The function that correctly depicts the scenario is;
f(x) = Value of buffet (1 + Rate of increment)^x
Substitute the values in the above equation, so;
f(x) = 18.50 (1 + 6%)^x
So, we have;
f(x) = 18.50 (1.06)^x
The buffet cost 10 years from now means that
x = 10
Substitute the values in the above equation,
f(10) = 18.50 (1.06)^10
Evaluate;
f(10) = 33.13
x = -15
Substitute the values in the above equation, so, we have
f(-15) = 18.50 (1.06)^(-15)
Evaluate;
f(-15) = 7.71
Hence, the amount 15 years ago is $7.71
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Find the exact length of the curve y=ln[(e^x+1)/(e^x-1)], a < x < b,
a > 0 (">"= greater than or equal to and "<" = less than or equal to)
The exact length of the curve y=ln[(e^x+1)/(e^x-1)], a ≤ x ≤ b, is equals to the ln[( eᵇ - e⁻ᵇ )/ ( eᵃ - e⁻ᵃ)].
The exact length of curves implies the length of arc of curve. Arc length is the distance between two points along a section of a curve, formula
Arc length = ₕ∫ᵏ√( 1 + ( y')² dx , h≤x≤k
We have a curve , y = ln[ (eˣ + 1) /(eˣ - 1) ] , a≤x≤b , a> 0
firstly, determine the first derivative of y with respect to x ..
dy/dx = y' = [ 1/(eˣ +1) /(eˣ -1)] d/dx (( eˣ + 1) /(eˣ -1) )
( apply chain rule )
= (eˣ -1 )/(eˣ +1 )[{(eˣ -1) eˣ - ( eˣ +1)eˣ}/(eˣ - 1)²]
( using quotient rule)
= (eˣ -1 )/(eˣ +1 )[(e²ˣ - eˣ - e²ˣ - eˣ)/(eˣ - 1)²]
= (eˣ -1 )/(eˣ +1 )[(- 2eˣ)/(eˣ - 1)²]
= -2eˣ /(eˣ -1 )(eˣ +1 )
dy/dx = - 2eˣ /e²ˣ -1
so, (dy/dx)² = (- 2eˣ /(e²ˣ -1) )²
= 4e²ˣ /(e²ˣ -1 )²
and 1 + (dy/dx)² = 1 + {4e²ˣ /(e²ˣ -1 )² }
= {(e²ˣ -1 )² + 4e²ˣ}/(e²ˣ -1 )²
= (e⁴ˣ + 1 - 2e²ˣ + 4e²ˣ)/(e²ˣ -1 )²
= ( e⁴ˣ + 1 + 2e²ˣ )/(e²ˣ -1 )²
1 + (dy/dx)² = (e²ˣ + 1)²/(e²ˣ -1 )²
Now plugging the value in arc length formula,
Arc length = ₐ∫ᵇ √{1 + (dy/dx)²} dx
= ₐ∫ᵇ √{(e²ˣ + 1)²/(e²ˣ -1 )²} dx
= ₐ∫ᵇ√{(e²ˣ + 1)/(e²ˣ -1 )}² dx
= ₐ∫ᵇ {(e²ˣ + 1)/(e²ˣ -1 )} dx
= ₐ∫ᵇ[eˣ(eˣ + e⁻ˣ)/eˣ(eˣ - e⁻ˣ )] dx
= ₐ∫ᵇ[(eˣ + e⁻ˣ)/(eˣ - e⁻ˣ )] dx
Arc length= ₐ∫ᵇ(cosh x)/(sinh x)dx ( using hyperbolic functions formulas )
let sinh x = t => cosh xdx = dt
and when x = a => t = sinh a
when x = b => t = sinh b
so, arc length = ₛᵢₙₕ ₐ∫ˢᶦⁿʰ ᵇ (1/t) dt
= [ ln( sinh b ) - ln(sinh a) ]
= ln( sinh b/ sinh a)
Arc length= [( eᵇ - e⁻ᵇ )/ ( eᵃ - e⁻ᵃ)]
Hence, length of curve is ln[( eᵇ - e⁻ᵇ )/ ( eᵃ - e⁻ᵃ)].
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What is nth degree polynomial?
A polynomial of the form ax^n + bx^(n-1) + cx^(n-2) + ... + z is called an nth-degree polynomial, where n is a positive integer. The highest exponent in the polynomial is n.
The term "nth-degree polynomial" refers to the highest exponent in the polynomial. In other words, a polynomial is an nth-degree polynomial if the highest exponent in the polynomial is n.
For example, the polynomial x^2 + 3x + 5 is a 2nd-degree polynomial because the highest exponent is 2. Similarly, the polynomial 3x^4 + 2x^3 - 5x^2 - 6x + 9 is a 4th-degree polynomial because the highest exponent is 4.
The degree of a polynomial can be helpful in determining the number of roots that the polynomial has. For example, a 2nd-degree polynomial will have at most 2 roots, while a 4th-degree polynomial can have up to 4 roots.
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The United States House of Representatives has 35 more members than four times the number of members in the United States Senate. Define a variable and write an expression to represent the number of members in the House of Representatives.
The expression to represent the number of members in the House of Representatives is 4x + 35.
How to calculate the expression?Based on the information, the question has to do with an expression. An expression shows the relationship between the variables.
In this situation, the United States House of Representatives has 35 more members than four times the number of members in the United States Senate
Let the number of members in Senate be x.
The expression will be:
(4 × x) + 35
= 4x + 35
The expression is 4x + 35.
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In a hypothesis test, if the computed p-value is less than 0.001,there is very strong evidence to a).fail to reject the null hypothesis. b).retest with a different sample. c). reject the null hypothesis.
The p-value of the hypothesis test is a measure of the strength of the evidence. There is significant evidence to reject the null hypothesis if it is less than 0.001.
A hypothesis test is a statistical procedure used to make a decision about an underlying population based on sample data. The hypothesis test involves two hypotheses. The null hypothesis, which states that the population parameter is equal to a given value, and the alternative hypothesis, which states that the population parameter is not equal to the given value. The strength of the evidence is gauged by the p-value. A p-value of less than 0.001 indicates that the evidence is very strong, and that the null hypothesis should be rejected. This means that the data is not consistent with the null hypothesis, and the alternative hypothesis should be accepted. In other words, the population parameter is not equal to the given value.
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If the lease factor is given as 0. 00045 what interest rate is that equivalent to.
If the lease factor is given as 0. 00045 then the interest rate is that equivalent to 4.5%
The lease factor of 0.00045 is equivalent to an interest rate of 4.5%. This is because the lease factor is a measure of the interest rate a lessee pays on a lease.
It is calculated by dividing the interest rate by 2400. This means that the higher the lease factor, the higher the interest rate that the lessee pays.
For example, if the interest rate is 12%, then the lease factor would be 0.005.
In this case, the lessee would be paying an interest rate of 12%, or 0.005 as a lease factor.
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evaluate the integral. (use c for the constant of integration.) dx (ax)2 − b2 3/2
The integral evaluates the area under a curve by taking the antiderivative of the given function and evaluating the limits of the integral. The final result is (1/3) ax3 - (1/2) b2 ax + c.
∫ (ax)2 − b2 3/2 dx = (1/3) ax3 - (1/2)b2 ax + c
The integral evaluates the area under a curve. It is calculated by taking the antiderivative of the given function and then evaluating the limits of the integral. To find the antiderivative, we use the power rule to calculate the derivative of the function and then integrate the result. In this case, the function is (ax)2 - b2 3/2. Using the power rule, we find that the derivative of this function is (3/2) ax2 - (1/2) b2 ax. We then integrate this result and add the constant of integration (c) to obtain the integral. The final result for this integral is (1/3) ax3 - (1/2) b2 ax + c.
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