The value of the definite integral ∫[tex]x????"(x)????(x) dx[/tex] on the interval [tex][1,4][/tex] is -5.
What do you mean by integration?Integration is a fundamental concept in calculus that deals with finding the area under a curve and calculating the accumulation of values over an interval. Integration can be thought of as the reverse of differentiation, which is the process of finding the rate of change of a function at a given point.
There are two main types of integration: definite and indefinite integration. Definite integration involves finding the exact area between a curve and the x-axis over a specified interval. Indefinite integration involves finding the general antiderivative of a function, which is a function that, when differentiated, will yield the original function.
The process of integration can be represented symbolically as ∫ (the integral symbol) and the result of an integration is typically represented as an antiderivative. The fundamental theorem of calculus states that differentiation and integration are inverse operations, so the derivative of an antiderivative is the original function.
Given that ????(1) = 8, ????(4) = 7, ???? '(1) = 9, and ???? '(4) = -3, we can use the method of integrating by parts to evaluate the definite integral ∫[tex]x????"(x)????(x) dx[/tex] on the interval [1, 4].
Let u = ????(x) and dv = x dx. Then du = ???? '(x) dx and v = x^2/2.
Using the method of integrating by parts, we have:
∫[tex]x????"(x)????(x) dx= x????(x)[/tex] ×[tex]\frac{x}{2} -[/tex]∫[tex](\frac{x^{2} }{2} )????'(x) dx[/tex]
We can use the same method of integrating by parts to evaluate the second integral on the right-hand side. Let [tex]u=\frac{x^{2} }{2}[/tex] and [tex]dv=????'(x) dx[/tex]. Then [tex]du=x dx[/tex] and [tex]v=????(x)[/tex].
So we have:
∫[tex]x????"(x)????(x) dx = x????(x)[/tex]× [tex]\frac{x}{2} - \frac{x}{2} ^{2}????(x) + c[/tex]
Where C is an arbitrary constant of integration.
We can use the given values of ????(1) and ????(4) to evaluate the definite integral:
∫[tex]x????"(x)????(x) dx[/tex] = 4????(4) * 4/2 - (4^2/2)????(4) + 4????(1) * 1/2 - (1^2/2)????(1)
= 28 - 14 * 7/2 + 4 * 8/2 = 28 - 49 + 16 = -5.
Therefore, the value of the definite integral ∫[tex]x????"(x)????(x) dx[/tex] on the interval [1, 4] is -5.
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-5q=8=r
what is the dependent variable and what is the independent variable.
Answer:
q is independent var, r is dependent
Step-by-step explanation:
6.) Vanilla Ice takes out an ordinary loan to pay back a rental company. The bank has a 10.3% ordinary
interest rate for 240 days. What is the principal that Vanilla Ice borrowed if the result is a $330.00 interest
charge?
Let P be the principal Vanilla Ice borrowed. The interest charge is calculated as P * 10.3% * 240/365.
So, 330 = P * 10.3% * 240/365
Solving for P, we get:
P = 330 / (10.3% * 240/365) = $3,116.71
Therefore, Vanilla Ice borrowed a principal of $3,116.71.
Simple Interest (SI) is calculated based on the principal amount only, and the interest rate stays constant throughout the tenure. The formula for Simple Interest is:
SI = P * R * T
Where:
P = Principal amount
R = Interest rate
T = Time (in years)
Compound Interest (CI) is calculated based on the principal amount and the interest earned on it in previous periods. The interest earned in each period is added to the principal, and the new amount becomes the principal for the next period. This leads to an exponential growth of the invested amount. The formula for Compound Interest is:
CI = P * (1 + R/n)^(nt)
Where:
P = Principal amount
R = Interest rate
n = Number of compounding periods per year
t = Time (in years)
In summary, the main difference between simple and compound interest is that simple interest is calculated only on the principal amount, whereas compound interest is calculated on the principal amount and the accumulated interest.
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f(x)=2x+3, g(x)=-x^2+5, (f \circ g)(x)
(f ◦ g)(x) = -2x^2 + 11.
What is Composition of two functions ?
The new function obtained by performing f first and then g is the combination of two functions g and f.
The composition of two functions f and g, denoted by (f ◦ g)(x), is defined as the function that maps x to f(g(x)).
Given the functions f(x) = 2x + 3 and g(x) = -x^2 + 5, the composition of these two functions is:
(f ◦ g)(x) = f(g(x)) = f(-x^2 + 5) = 2(-x^2 + 5) + 3 = -2x^2 + 11
So, (f ◦ g)(x) = -2x^2 + 11.
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in a party, 5 male-female couples attend. the males and females are randomly paired for a dance. what is the probability that exactly two couples are dancing together?
The probability of exactly 2 couples dancing together is 100/45 = 20/9.
To find the probability of the total number of ways to form pairs of 5 couples is C(10, 2) = 45, because there are 10 people and we need to choose 2 at a time.
The number of ways to form exactly 2 couples dancing together is C(5, 2) * C(5, 2) = 10 * 10 = 100. This is because we first choose 2 couples out of the 5 couples and then pair the males and females within each chosen couple.
So the probability of exactly 2 couples dancing together is 100/45 = 20/9.
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The probability of exactly 2 couples dancing together is 100/45 = 20/9.
To find the probability of the total number of ways to form pairs of 5 couples is C(10, 2) = 45, because there are 10 people and we need to choose 2 at a time.
The number of ways to form exactly 2 couples dancing together is C(5, 2) * C(5, 2) = 10 * 10 = 100. This is because we first choose 2 couples out of the 5 couples and then pair the males and females within each chosen couple.
So the probability of exactly 2 couples dancing together is 100/45 = 20/9.
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What is the numeric coefficient for 3x4y2
Answer: The numeric coefficient is 3.
Step-by-step explanation:
The equation of Line D is y = 3/8x +3. The equation of the line E is y = 3/8x + 8/3, Are line D and Line E perpendicular,parallel or neither.
The slope of line D and line E will be congruent. Then the lines are parallel to each other.
What is a linear equation?A connection between a number of variables results in a linear model when a graph is displayed. The variable will have a degree of one.
The linear equation is given as,
y = mx + c
Where m is the slope of the line and c is the y-intercept of the line.
If the slopes are different then the lines are intersecting.If the slopes are equal but have different intercepts then the lines are parallel.If the slopes and intercepts are equal then the lines are coincident.If the product of the slope is negative then the lines are perpendicular.The equations are given below.
Line D: y = (3/8)x +3
Line E: y = (3/8)x + 8/3
The slope of line D and line E will be congruent. Then the lines are parallel to each other.
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What is the equation in standard form of the line that passes through the point (6, 1) and is parallel to the line represented by 8x + 3y= 15?.
The equation in standard form of the line that passes through the point (6, 1) and is parallel to the line represented by 8x + 3y= 15 is equal to y = -8x + 43.
What is the point-slope form?Mathematically, the point-slope form of a straight line can be calculated by using this mathematical expression:
y - y₁ = m(x - x₁) or y - y₁ = (y₂ - y₁)/(x₂ - x₁)(x - x₁)
Where:
m represents the slope.x and y represents the data points.From the information provided above, we have the following slope and data points on the line:
Points = (6, 1).Slope, m = -8.In Geometry, two (2) lines are parallel under the following conditions:
m₁ = m₂ ⇒ -8 = -8
At data point (6, 1), a linear equation of this line can be calculated in point-slope form as follows:
y - 1 = -8(x - 6)
y - 1 = -8x + 42
y = -8x + 42 + 1
y = -8x + 43
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Select the correct answer. Which inequality is equivalent to the given inequality? -4(x+7)< 3(x-2)
The inequality is equivalent to the given inequality? -4(x+7)< 3(x-2) is x>-22/7
How to find the inequality?Inequality is a mathematical statement which shsow that two quantities are not the same or that they are not equal.
The given inequality is -4(x+7) < 3(x-2)
Opening the brackets to have
-4x-28<3x-6
Collecting like terms to have
-4x-3x<-6+28
This shows that -7x<22
Making x the subject of the relation we have
x>-22/7
The inequality that satisfies the given inequality is x>-22/7
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Can anybody understand this please help
Trey is 4 times older than his sister Kelli. If K stands for Kelli's age, write an expression to represent Trey's age . (help please lol)
Answer:
The expression is:
4K
But as an equation, it would be:
T = 4K
Step-by-step explanation:
K = Kelli's age
T = Trey's age
Trey is 4 times older than Kelli
T = (4)(K)
What is the quotient of 11/14 divided by 11/21
Answer: 1.5
Step-by-step explanation:
How many ways can you split 12 people into 3 teams of 4?
By using permutations and combinations, We can arrange 12 people into 3 teams of 4 in 5775 different ways.
Here, we will use the concept of permutations and combinations to solve the question.
We have to arrange 12 people into 3 teams of 4.
We know that ⁿCr = n! / [(n - r)! × r!]
No. of ways to select the first 4 people in the first group = ¹²C₄
= 12! / [(12 - 4)! × 4!]
= 12! / [(8! × 4!)]
= 495
No. of ways to select 4 people from the remaining 8 for the second group = ⁸C₄ = 8! / [(8 - 4)! × 4!] = 8! / [(4! × 4!)] = 70
No. of ways to select 4 people from 4 for third group = ⁴C₄ = 4! / [(4 - 4)!] × 4!] = 4! / [(0! ×4!)] = 1
Total no. of ways to select people for group = (495 x 70 x 1) / 3! = 34650 /6 = 5775
Hence, we can split 12 people into 3 teams of 4 in 5775 different ways.
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Find the following for each function.
Domain:
Relative Min:
Range:
Increasing:
Relative Max:
Decreasing:
For the given function; Domain: -4 ≤ x ≤ 4, Range: 0 ≤ y ≤ 3, Relative max: x = 0, y=3, and Relative min: x = -2, y=0 and x = 2, y =0.
What is domain and range?The range of values that we are permitted to enter into our function is known as the domain of a function.
The x values for a function like f make up this set (x).
A function's range is the collection of values it can take as input. The values in this set
Hence, once we input an x value, the function returns. The y values are those.
The domain of the function is the input values, that is the x-coordinates thus,
Domain: -4 ≤ x ≤ 4
Range of the function is the output values, that is the y-coordinate thus,
Range: 0 ≤ y ≤ 3
Relative max is a point where the function changes from increasing to decreasing thus,
Relative max: x = 0, y=3
Relative min is a point where the function changes from decreasing to increasing, thus,
Relative min: x = -2, y=0 and x = 2, y =0
Hence, the Domain: -4 ≤ x ≤ 4, Range: 0 ≤ y ≤ 3, Relative max: x = 0, y=3, and Relative min: x = -2, y=0 and x = 2, y =0.
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Mainsail
Headsail
Welcome to the land down under! Spend the day sailing
in Sydney Harbor! Sailboats are made up of two sails,
the mainsail and the headsail. Typically, these sails are
the shape of triangles and these two triangles are
similar. Considering the sailboat pictured below,
determine the value of x and y and the unknown side
lengths.
6+A
X-2
30 ft
5y-3
26 ft
By using the properties of similar triangles, the values of x and y are respectively, 15 & 7
What is the similarity of triangles?Triangles with the same shape but different sizes are said to be similar triangles. Squares with any side length and all equilateral triangles are examples of related objects. In other words, if two triangles are similar, their corresponding sides are proportionately equal and their corresponding angles are congruent. Triangle resemblance is shown here by the symbol "≈"
Given that,
Side lengths of triangle 1, y + 9, x, x -2
Side lengths of triangle 2, 30, 26, 5y - 3
Both the triangles are similar,
So, we can write
(x - 2)/x = 26/30
After solving it, x = 15
Similarly,
(y + 9)/x = (5y - 3)/30
y = 7
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price of a stock over a 5-day period.
Closing Price
$104.67
$103.95
$109.35
$108.27
$113.58
Day
Monday
Tuesday
Wednesday
Thursday
Friday
What is the average of the 3-day simple moving averages for these five days to the nearest $0.01?
The average of the 3-day simple moving averages for these five days to the nearest $0.01 is $105.99, $107.19 and $110.40.
What is a Simple Moving Average?
By dividing the number of periods inside the range by the average of the range of prices, simple moving averages are used.
We are given closing prices of a stock over a 5-day period.
So, 3-day simple moving averages are
For Monday - $104.67
For Tuesday - $103.95
For Wednesday - ($104.67 + $103.95 + $109.35)/3 = $105.99
For Thursday - ($103.95 + $109.35 +$108.27)/3 = $107.19
For Friday - ($113.58 + $109.35 +$108.27)/3 = $110.4
Hence, the average of the 3-day simple moving averages for these five days to the nearest $0.01 is $105.99, $107.19 and $110.40.
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Matt is using 2 mixtures of saltwater solution to make a 6% saltwater solution for his aquarium. One mixture is a 2% saltwater solution, and the other is an 8% saltwater solution. Matt made a chart to calculate the correct amount of each solution to use. Using the information in the table, which equation can Matt use to find x, the amount of 2% mixture he needs to use for his aquarium?
Answer:
matt needs to use more than 2% mixture for his aquarium
Step-by-step explanation:
Jason walks a 5-mile scenic walkway that stretches from the west to the east end of the park. There is a distance marker at each
mile and one at the east end of the walkway.
What is the total number of markers along the walkway?
Answer:
The total number of markers along the 5-mile walkway is 6.
This includes markers at each mile (5 markers) and one marker at the east end of the walkway.
i asked my class what they had for breakfast. 10 students had cereal, 5 students had eggs, and 3 students had no breakfast. what kind of data do i have?
The kind of data we have is Categorical data. Data that can be broken down into groups or categories is known as categorical data.
There are three categories in this case: eggs, cereal, and no breakfast These groups include the 10 students who ate cereal, 5 students who ate eggs, and 3 students who did not eat breakfast at all.
A categorical variable is a variable in statistics that can have one of a limited, typically fixed number of possible values and assign each observational unit to a specific group or nominal category based on some qualitative property.
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solve Edson exercises in sets that are each t minutes long. He does 6 sets of push-ups, 3 sets of pull-ups, and 4 sets of sit-ups. Use an expression to represent the time it takes Edson to exercise as the sum of three different terms. Simplify the expression. Enter your answer in the box.
The time taken by Edson to complete his workout routine is 13 t
What is the unitary method?The unitary method is a method in which you find the value of a unit and then the value of a required number of units.
Given here: Edson exercises in sets that are each t minutes long. He does 6 sets of push-ups, 3 sets of pull-ups, and 4 sets of sit-ups
Now Edson does 6 sets of pushups time taken to finish the sets are
Time ( pushups)=6t
similarly, the time it takes to complete the other exercises are
Time(pull ups)=3t
Time(sit ups)=4t
Thus total time taken is equal to 3t+4t+6t=13t
Hence, The time taken by Edson to complete his workout routine is 13 t
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7. Josh takes a twenty-question multiple-choice exam where each question has five possible answers. Some of the answers he knows, while others he gets right just by making lucky guesses. Suppose that the conditional probability of his knowing the answer to a randomly selected question given that he got it right is 0. 92. How many of the twenty questions was he prepared for
The total number of twenty questions prepared by Josh can be find by using the expression k = 20 * P(getting a question right) / (1 + 20 * 0.08 / 0.92), where, k = total question prepared by Josh
Let's call the number of questions Josh was prepared for as "k".
The probability of Josh getting a question right given that he knew the answer is 0.92,
So the probability of him getting a question right by guessing is
1 - 0.92 = 0.08.
The overall probability of Josh getting a question right is the sum of the probability of him getting it right given that he was prepared and the probability of him getting it right by guessing:
P(getting a question right) = P(getting a question right | prepared) * P(prepared) + P(getting a question right | guessed) * P(guessed)
Since each question is either prepared or guessed, the sum of P(prepared) and P(guessed) is 1:
P(getting a question right) = 0.92 * P(prepared) + 0.08 * (1 - P(prepared))
Simplifying the expression:
P(getting a question right) = 0.92 * P(prepared) + 0.08
Since the overall probability of Josh getting a question right is the same for every question,
it must be the same for all twenty questions:
So, we can write it as:
0.92 * P(prepared) + 0.08
= 0.92 * k/20 + 0.08 * (20-k)/20
Dividing both sides by 0.92:
P(prepared) = k/20 = (0.92 * k + 0.08 * 20 - 0.08) / 0.92
Isolating k:
k = 20 * P(prepared) / (1 + 20 * 0.08 / 0.92)
Plugging in the overall probability of Josh getting a question right, we have:
k = 20 * P(getting a question right) / (1 + 20 * 0.08 / 0.92)
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4Y - 1 when y = -5 PLEASE HELP MEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE
Answer: -21
Step-by-step explanation: first you multiply 4 x -5 or 4(-5) - 1 4 x -5 would give you -20 and -20 - 1, since both are negative we add to get a total of -21 hope this helps.
Which is the independent variable and
which is the dependent variable?
The number of dollar bills, d, is the
?
independent
dependent
variable.
arters, q, is the
every bill she puts in.
variable.
Answer:The number of dollar bills, d, is the independent variable.
The quarters, q, is the dependent variable.
Step-by-step explanation:
The number of dollar bills, d, is the independent variable.
The quarters, q, is the dependent variable.
Answer:
Step-by-step explanation:
dependent then independent because i passes the lesson.
find the limit l (if it exists). if it does not exist, explain why. (if an answer does not exist, enter dne.) lim x→49 x − 7x − 49
The limit does exist for lim x→49 (x^2 - 7x - 49) and it approaches value of 2009.
The limit can be found using algebraic manipulation. First we will try plug in the value and see if we can determine the limit. We have:
lim x→49 (x^2 - 7x - 49) = 49^2 - 7×49 - 49 = 2009
So limit exist at x^2 - 7x - 49 and it can be found out by simple substitution.
So, as x approaches 49, the expression x^2 - 7x - 49 approaches 2009.
Therefore, lim x→49 x^2 - 7x - 49 = 343.
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Write the decimal expansion for 5 9. ( 7Th GRADE)! HURRY HELPPPP…
Answer: 0.56
Step-by-step explanation: convert from fraction to decimal
Given the four points E(2,5), F(7,1), G(2,-3), and H(-8,5) is EF parallel to GH?
EF is parallel to GH by using the slope formula. Both the slope is equal to -4/5.
Given that the information below:
E(2,5), F(7,1), G(2.-3) and H(-8,5) is EF parallel to line GH.
By using slope formula method we can compare the slopes of line EF & line GH.
Where m = (y2 - y1 ) / (x2 - x1) slope of segment become (x1,y1) & (x2,y2)
EF Slope = (1-5)/(7-2) = -4/5
GH Slope = (5-(-3)/(-8-2) = 8/-10 = -4/5
By using slope formula method EF slope is -4/5 & GH slope is -4/5. Both the slopes of the equation is equal. In this way we can prove that the EF is parallel to GH.
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on+may+11,+corlis+took+out+a+loan+for+$2,900,+at+8%+ordinary+interest,+for+82+days.+what+is+the+maturity+date+of+the+loan?
The maturity date of the loan is July 31st.
To find the maturity date, you first need to find the number of days between May 11th and the maturity date. The loan was taken out for 82 days, so you add 82 days to May 11th to find the maturity date.
For example, if May 11th is day 1, then day 82 would be July 31st. Therefore, the maturity date of the loan is July 31st.
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what is the slope of a line that has the coodinates of (10,23) & (20,36)?
Step-by-step explanation:
The slope of a line can be calculated as the difference of the y-coordinates divided by the difference of the x-coordinates of two points on that line. In this case, we have two points: (10,23) and (20,36).
So, the slope of the line would be:
(36 - 23) / (20 - 10) = 13 / 10 = 1.3.
Therefore, the slope of the line that has the coordinates of (10,23) and (20,36) is 1.3
HELPP PLEASEE I NEED TO KNOW THE METHOD AND Determine if the triangles are similar if similar write a similarity statement that explain which method used
Triangle ABC is similar to triangle DEF because their corresponding angles, ∠A and ∠D, and ∠B and ∠E, are equal.
The AAA similarity is a theorem of a triangle which means "angle-angle-angle", that is, two triangles are similar to each other if they both have three corresponding angles that are congruent to each other.
The line BC divides the line AD and AE, we can state that the two triangles are similar by the Angle-Angle-Angle (AAA) Similarity Theorem.
The AAA Similarity Theorem states that if the three angles of one triangle are equal to the three angles of another triangle, then the two triangles are similar.
Here is an example similarity statement based on the Angle-Angle-Angle (AAA) Similarity Theorem: "Triangle ABC is similar to triangle DEF because they both have equal corresponding angles, ∠A and ∠D, ∠B and ∠E, and ∠C and ∠F, and because the line BC divides the line AD and AE."
Hence, Triangle ABC is similar to triangle DEF because their corresponding angles, ∠A and ∠D, and ∠B and ∠E, are equal.
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How do you convert 50 Kilogram (kg) to Pound (lb)?
Multiply the mass in kilogrammes by 2.2046226218 to convert 50 kg to pounds.
What makes anything a lb?Libra pondo, which meaning "a pound by weight," was the unit of measurement in ancient Rome when the word "pound" first appeared. According to the BBC, the pondo portion of the sentence is where the English term "pound" originates. However, the word's libra portion is where the abbreviation "lb" comes from.Follow these easy procedures to convert a weight or mass measurement from kilogrammes (kg) to pounds (lbs): Add the conversion factor 2.2046 to the weight. Therefore, 50 kg would be equal to 110.23 kg when multiplied by 2.2046.Multiply the mass in kilogrammes by 2.2046226218 to convert 50 kg to pounds. [lb] = 50 * 2.2046226218 is the formula to convert 50 kilogrammes to lbs.To learn more about Kilogram refer to:
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which of the following is an antiderivative of f(x)=tan(ex π2) on 0≤x≤ln(π2) ?
The antiderivative of f(x) = tan(eˣ + π/2) on 0≤x≤ln(π/2) is:
⇒ ln |sec([tex]e^{\pi /2}[/tex])| / [tex]e^{\pi /2}[/tex]
Now, For the antiderivative of f(x) = tan(eˣ + π/2) on the interval 0≤x≤ln(π/2), we need to integrate the given function.
Now, using the substitution u = eˣ + π/2.
Then, we have:
du/dx = eˣ
dx = du / eˣ
Using this substitution, we can rewrite the original function as:
f(x) = tan(eˣ + π/2) dx = tan(u) du / eˣ
Now let's integrate this function:
∫tan(u) du / eˣ = - ln |cosec(u)| / eˣ + C
Substituting back u = eˣ+π/2:
ln |cosec(eˣ + π/2)| / eˣ + C
= - ln |cosec(eˣ + π/2)| / eˣ + C
Now we need to evaluate this expression at the limits of integration x = 0 and x = ln(π/2):
F(ln(π/2)) - F(0)
= - ln |cosec([tex]e^{\pi /2 } + \frac{\pi }{2}[/tex])| / [tex]e^{\pi /2}[/tex] + C - (- ln |cos(0)| / [tex]e^{\pi /2}[/tex] + C)
= - ln |sec([tex]e^{\pi /2}[/tex])| / [tex]e^{\pi /2}[/tex] + ln |cos(0)| / [tex]e^{\pi /2}[/tex]
= - ln |sec([tex]e^{\pi /2}[/tex])| / [tex]e^{\pi /2}[/tex] + ln(1) / [tex]e^{\pi /2}[/tex]
= ln |sec([tex]e^{\pi /2}[/tex])| / [tex]e^{\pi /2}[/tex]
So, the antiderivative of f(x) = tan(eˣ + π/2) on 0≤x≤ln(π/2) is:
⇒ ln |sec([tex]e^{\pi /2}[/tex])| / [tex]e^{\pi /2}[/tex]
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