If the calculated Cp and Cpk values are greater than 1, the process is considered capable. If Pp and Ppk are greater than 1, the process is considered capable.
To calculate Cp, Cpk, Pp, and Ppk using MINITAB, you would need to input the data into the software and run the appropriate statistical analysis.
Cp and Cpk are measures of a process's capability, they are calculated as follows:
Cp = (USL - LSL) / (6*σ)Cpk = min(Cpu, Cpl)Where,
USL and LSL are the upper and lower specification limitsσ is the standard deviation of the process.Pp and Ppk are similar to Cp and Cpk, but they are calculated using the range of the data instead of the standard deviation.
Pp = (USL - LSL) / (4*R)Ppk = min(Ppu, Ppl)Where R is the range of the data.
If the calculated Cp and Cpk values are greater than 1, the process is considered capable. If Pp and Ppk are greater than 1, the process is considered capable.
It is worth noting that Capability analysis can only be performed on stable and in-control processes. If the process is not in control, analyzing process capability is not meaningful.
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What are the dimensions of the product?
[2 1 0] x [1 -1 2 -1 -2 1 0 1 1] =
The resulting vector has the same dimensions as the original, 1 x 3.
What is a matrix?A rectangular array of numbers, symbols, or expressions set up in rows and columns is known as a matrix. They are employed in a wide variety of mathematical disciplines and contexts, including calculus, statistics, physics, and engineering, as well as linear algebra. Matrices are an essential tool for solving systems of linear equations and representing linear transformations since they may be added, subtracted from, and multiplied by other matrices.
The matrix multiplication of [2 1 0] and [1 -1 2 -1 -2 1 0 1 1] is:
[2 1 0] x [1 -1 2 -1 -2 1 0 1 1] = [2(1) + 1(-1) + 0(2), 2(-1) + 1(-2) + 0(1), 2(0) + 1(1) + 0(1)]
= [2 -3 1]
Therefore, the product of the two matrices is a row vector [2 -3 1], which has the dimensions 1 x 3.
If we subtract 1 from each element of this row vector, we get:
[2 -3 1] - 1 = [1 -4 0]
So, the resulting vector has the same dimensions as the original, 1 x 3.
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Which calculation shows the best method for estimating the result of 384 divided by three-sixteenths?
400 divided by one-fourth equals 1,600
400 divided by one-fourth equals 100
300 divided by one-third equals 900
300 divided by one-third equals 100
Answer:
Step-by-step explanation:
Select the correct choice that completes the sentence below. An odd function is symmetric with respect to the y-axis. origin line y =X-axis.
From the given option , the completed sentence is : An odd function is symmetric with respect to the origin .
The Odd Functions are functions which return the negative inverse when x is replaced with "-x" . which means that function f(x) is an odd function when f(-x) = -f(x) .
The Odd Functions are symmetrical about origin because ; the Odd function on one side of x-axis is sign inverted with respect to the other side or graphically, symmetric about the origin.
Therefore , the odd functions are symmetric about the origin .
The given question is incomplete , the complete question is
Select the correct choice that completes the sentence below.
An odd function is symmetric with respect to the _____ .
(a) y-axis
(b) origin
(c) line y = x
(d ) x-axis.
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Maria will spend less than $29 on gifts. So far, she has spent $17. What are the possible additional amounts she will spend?
Use c for the additional amount (in dollars) Maria will spend.
Write your answer as an inequality solved for c.
Answer:
Maria is able to spend less than $12.
C < 12
Step-by-step explanation:
If Maria has already spent 17 dollars, that means she has 12 dollars left. C cannot be greater than 12. It has to be less.
So think of this C < $12
So if we do the total on one side and the $17 and C on the other side of inequality we get
C + 17 < 29
It cannot be equal to 29 so that's why we don't use a less than or equal to symbol.
Simplify this to get C < 12
I hope this is able to help you :D
Mrs. Barton's famous peanut butter cookies call for 1 cup of peanut butter for every
1
2
of a cup of oil. Today, she wants to make a huge batch with 1 cup of oil. How much peanut butter should she use?
Mrs. Barton should use 2 cups of peanut butter in her huge batch with 1 cup of oil.
What is fraction?
Fractions are a way of representing part of a whole. They are written as two numbers separated by a slash. The top number is called the numerator, and the bottom number is called the denominator. For example, in the fraction 1/2, 1 is the numerator and 2 is the denominator. This fraction represents one part of a total of two parts.
Mrs. Carter's famous peanut butter cookies call for 1 cup of peanut butter for every 1/2 of a cup of oil. Today, she wants to make a huge batch with 1 cup of oil.
From the above question:
1/2 cup of oil = 1 cup of peanut butter
1 cup of oil = x
Cross Multiply
1/2x = 1 × 1
x = 1 ÷ 1/2
x = 1 × 2/1
x = 2 cups of peanut butter.
Therefore, for huge batch with 1 cup of oil, she should use 2 cups of peanut butter.
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Ryan claims that the cost of Plan A and Plan B differ a lot.
Which statement best explains why Ryan was misled by the graph?
a.) The two graphs use different vertical scales.
b.) The scale of intervals on the horizontal axis of both graphs
is inconsistent.
c.) The graph of Plan B shows a rapid increase in price.
E
d.) The graph of Plan A shows a gradual increase in price.
IfIf Ryan claims that the cost of Plan A and Plan B differ a lot. The statement that best explains why Ryan was misled by the graph is: b.) The scale of intervals on the horizontal axis of both graphs is inconsistent.
The reason Ryan was misled by the graph?Based on the graph we can see that the graph is inconsistent based on the fact that the scale of intervals on the horizontal axis of Plan A graph and Plan B graph are inconsistent.
The reasons why the both graphs are inconsistent was because the lines did not intersect or are not on the same line which means that they are parallel lines.
Therefore the correct option is B.
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Answer:
Step-by-step explanation:
The two graphs use different vertical scales.
The sum of the differences between sample observations and the sample mean is
a. Zero
b. The range
c. The mean deviation
d. The standard deviation
The sum of the differences between sample observations and the sample mean is zero.
Statistics is a science that deals with methods of collecting data, processing data, presenting data, analyzing data, drawing conclusions, and making reasonable decisions based on existing facts.
There are some terms that are frequently used in statistics:
Range is the difference between the largest and the smallest values in a data set;Mean is the average of the sample values;Mean deviation is the average deviation of the data from the mean;Variance is a measure of how far the data is spread from its average value, which is equal to the average distances of the sample observations from the sample mean;Standard deviation is a value used to determine the distribution of data in a sample and see how close the data is to the mean value. The standard deviation is equal to the square root value of the variance.The sum of the differences between sample observations and the sample mean can be denoted as ∑(xi - mean), where n is the size of the sample and i = 1,2,3,...,n
∑(xi - mean) = ∑xi - n*mean,
= n*mean - n*mean
= 0
Thus the sum of the differences between sample observations and the sample mean is zero.
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The three interior angles of a triangle measure x°, (x-4)°, and (x+16)°. Which is the measure of one of the three angles?
OA) 48°
O B) 52°
O C) 60°
OD) 70°
=======================================================
Explanation:
Every triangle has the rule where the inside angles add to 180.
(angle1)+(angle2)+(angle3) = 180
x + (x-4) + (x+16) = 180
3x+12 = 180
3x = 180-12
3x = 168
x = 168/3
x = 56
Then we can determine each angle.
angle1 = x = 56 degreesangle2 = x-4 = 56-4 = 52 degrees (choice B)angle3 = x+16 = 56+16 = 72 degreesAs a check:
angle1+angle2+angle3 = 56+52+72 = 180
This confirms the answer is correct.
Write the following ratio using two other notations.
6:5
Use only the numbers above (not any others).
Notation one:
Notation two:
0.
3
0=0
0:00 to
0<0 0>
The ratio may be written if it is a component of a percentage.
6 is to 5 is 12 is to 10 .
What is a ratio notation?Ratios can be expressed as a notation, like 1/2, or as a ratio, like 6:5. A ratio's value is determined by dividing the first and second numbers. Consider the ratio 12:10 as an example. "6 to 5" should be read as the ratio. Any fraction, like 6/5, can also be expressed as a ratio, which is written as 6:5.
A colon or a fraction bar are two typical symbols representing ratios. You can also use words.
6 : 5 = 6/5 = 6 to 5
The ratio may be written if it is a component of a percentage.
6 is to 5
as in ...
12 is to 10 what 6 is to 5, and vice versa.
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Find IJ and JK in parallelogram HIJK.
h 6p
i 7p+17
j 9p+11
K
The values of IJ and JK in parallelogram HIJK are 52 and 80 respectively
How to find the values?Recall that in every parallelogram, the opposite sides are equal and parallel.
This implies that IJ ║ HK and HI ║ JK
⇒ 5s - 88 = 3s - 32
Collecting the like terms we have
5s-3s = -32 +88
2s = 56
This implies that s = 28
By substitution we have s as 3(28) -32
= 84-32 = 52 and and 4(28)-32 = 80
The sides of the parallelogram are 80 and 52 respectively
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The number 1465 rounded to its highest place value is?
The table shows the balance of Rob’s checking account at the end of the day. This is $95.50 less than the amount he had the beginning of the day.
Which of the following equations can be used to determine Rob’s account balance at the beginning of the day?
Frank rented a unicycle at the circus. He paid a flat rental fee of $15 plus $2 and hour. The total cost was $23. Write and solve an equation you can use to find the number of hours he rented the unicycle.
Answer:
4 hrs
Step-by-step explanation:
let x = # of hours he rented the unicycle
$15 + ($2/hr)x = $23
($2/hr)x = $23 - $15 = $8
x = ($8) / ($2/hr) = 4 hours
A 54-foot long wire will be attached to the top of a pole for support, and will be pulled tight and anchored to the ground.
If the wire makes a 13 degree angle with the ground, how tall is the pole?
The height of the pole used in the question is; 12.15 ft
How to use trigonometric ratios?The 3 main trigonometric ratios are;
sin x = opposite/hypotenuse
cos x = adjacent/hypotenuse
tan x = opposite/adjacent
Now, since the guy wire makes an angle of 13 degrees with the ground, it means the guy wire will be the hypotenuse of the right angle triangle and the height of the pole which we will label as h will be the opposite side of the triangle. Thus;
height/54 = sin 13
height = 54 * sin 13
height = 54 * 0.225
Height = 12.15 ft
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NO LINKS!!!
12. Find the inverse of the following functions using what you know about inverse operations. Some of these functions do not have a functional inverse. If it has no functional inverse, write "No inverse function"
Answer:
a) No inverse function
[tex]\textsf{b)} \quad c^{-1}(x)=5+\sqrt{x-2},\;\; c\geq2[/tex]
[tex]\textsf{c)} \quad f^{-1}(x)=-\dfrac{2x}{3x-5}[/tex]
Step-by-step explanation:
Part aGiven function:
[tex]a(x)=-4x^2+3[/tex]
To find the inverse of the function, swap x and y:
[tex]\implies x=-4y^2+3[/tex]
Rearrange the equation to isolate y:
[tex]\implies x-3=-4y^2[/tex]
[tex]\implies y^2=-\dfrac{x-3}{4}[/tex]
[tex]\implies y^2=\dfrac{3-x}{4}[/tex]
As the domain and range of function a(x) are unrestricted, the inverse relation is a “sideways” parabola and therefore it is not a function, since it fails the vertical line test.
Part bGiven function:
[tex]c(x)=(x-5)^2+2,\;\;x\geq5[/tex]
As the domain of function c(x) is restricted to x ≥ 5, the range is also restricted.
Range: c(x) ≥ 2To find the inverse of the function, swap x and y:
[tex]\implies x=(y-5)^2+2[/tex]
Rearrange the equation to isolate y:
[tex]\implies x-2=(y-5)^2[/tex]
[tex]\implies \pm\sqrt{x-2}=y-5[/tex]
[tex]\implies y=5\pm\sqrt{x-2}[/tex]
As the domain of function c(x) is restricted to x ≥ 5 then:
[tex]\implies y=5+\sqrt{x-2}[/tex]
Replace y with c⁻¹(x):
[tex]\implies c^{-1}(x)=5+\sqrt{x-2}[/tex]
The domain of the inverse of a function is the same as the range of the original function. Therefore, the domain of the inverse function is restricted to x ≥ 2.
Part cGiven function:
[tex]f(x)=\dfrac{5x}{3x+2}[/tex]
To find the inverse of the function, swap x and y:
[tex]x=\dfrac{5y}{3y+2}[/tex]
Rearrange the equation to isolate y:
[tex]\implies x(3y+2)=5y[/tex]
[tex]\implies 3xy+2x=5y[/tex]
[tex]\implies 3xy-5y=-2x[/tex]
[tex]\implies y(3x-5)=-2x[/tex]
[tex]\implies y=-\dfrac{2x}{3x-5}[/tex]
Replace y with f⁻¹(x):
[tex]\implies f^{-1}(x)=-\dfrac{2x}{3x-5}[/tex]
Let U = {(q, r, s, t, u, v, w, x, y, z)
A = {q, s, u, w, y}
B = {q, s, y, z)
C = {v, w, x, y, z).
List the elements in the set.
AL(BNC)
A. {q, s, u, w, y, z)
B. q, r, w, y, z}
C. {q, y, z)
D. {q, w, y}
PLEASE HELP!
The elements of the set A'(B∩C)is { r, t , v, x, y, z}
How to determine the setIt is important to note that the universal set is the set containing all the elements for a given event.
From the information given, we have that;
U = {(q, r, s, t, u, v, w, x, y, z)A = {q, s, u, w, y}B = {q, s, y, z)C = {v, w, x, y, z)The signs used to represent sets are;
∩ - Intersection of sets used to represent the common elements of two or more sets∪- Union of sets used to represent the sum of sets without repetitionTo determine the set,
A'(B∩C)
A' = { r, t, v, x, z}
B∩C = { y, z}
Now, the set A'(B∩C)
A'(B∩C) = { r, t , v, x, y, z}
Hence, the set is { r, t , v, x, y, z}
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question content area top part 1 express the following angular speed in radians per second. 14 revolutions per second
The angular speed of 14 revolutions per second is equal to 87.96 radians per second.
Angular speed is a measure of how fast an object is rotating or revolving around an axis. It is typically measured in units of angular displacement per unit of time, such as degrees per second, revolutions per minute (RPM), or radians per second.
To convert angular speed from revolutions per second (RPS) to radians per second (RAD/s), we can use the following formula:
Angular speed (RAD/s) = 2π x Revolutions per second (RPS)
Given that the angular speed is 14 revolutions per second, we can substitute this value into the formula:
Angular speed (RAD/s) = 2π x 14 RPS
Angular speed (RAD/s) = 87.96 RAD/s
So, 14 revolutions per second is equivalent to 87.96 radians per second.
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what is the position of (L) with respect to (c)
By the method of logical reasoning and the seating positions of seven people it is obtained that F sits third to the left of C.
What is logical reasoning?
The technique of applying systematic, logical steps based on mathematical logic to solve a problem is known as logical reasoning. Based on the information provided and mathematical principles, inferences can be drawn.
This seating positing problem can be solved by logical reasoning.
There are seven people - A, B, C, D, E, F, G.
Firstly, A sits third from one of the extreme ends.
Then two people sit between A and B.
The positions can be -
__ __ A __ __ B __ or __ B __ __ A __ __
Now, F sits fourth to the left of B.
So, second seating position gets eliminated.
Also, C sits at the immediate left of B.
__ F A __ C B __
Now, G sits third to the right of E.
E does not sit at any of the extreme ends.
D F A E C B G
So, the seating arrangement is obtained.
Therefore, F sits third to the left of C.
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Seven people named A, B, C, D, E, F, and G are sitting in a row facing north but not necessarily in the same order. A sits third from one of the extreme ends. Two people sit between A and B. C sits at the immediate left of B. F sits fourth to the left of B. G sits third to the right of E. E does not sit any of the extreme ends.
What is the position of F with respect to C?
Select the correct answer from each drop-down menu.
Consider the function f(x) = |₂³| − 1.
The piecewise functions that coincide with the given function are
[(x + 3)-1, when x is greater than or equal to 3,
[(x-3)/2] -1, when x is less than 3,
[(x+3)/2] -1, when x is greater than -3,
[-(x+3)/2] -1, when x is greater than or equal to -3,
and
Yes, the piecewise functions that coincide with the given function are [(x + 3)-1, when x is greater than or equal to 3, [(x-3)/2] -1, when x is less than 3, [(x+3)/2] -1, when x is greater than -3, and -(x+3)/2] -1, when x is greater than or equal to -3.
Piecewise functionThe piecewise functions that coincide with the given function are as follows: when x is greater than or equal to 3, the piecewise function is (x + 3)-1; when x is less than 3, the piecewise function is (x-3)/2 -1; when x is greater than -3, the piecewise function is (x+3)/2 -1; and when x is greater than or equal to -3, the piecewise function is -(x+3)/2 -1.This means that, depending on the value of x, the resulting value of the function f(x) = |₂³| − 1 will be different.For example, when x is equal to 3, then f(x) = 0; when x is equal to -3, then f(x) = -2; and when x is equal to 1, then f(x) = -1.Thus, the piecewise functions correctly express the given function, as they correctly capture the different values of f(x) for different values of x.No, the piecewise functions that coincide with the given function are[(x + 3)-1, when x is greater than or equal to 3,[(x-3)/2] -1, when x is less than 3,[-(x-3)/2] -1, when x is less than or equal to -3,[(x-3)/2] -1, when x is greater than -3.To learn more about Piecewise function refer to:
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Question 6.
Jim is studying growth of underwater plants in an aquarium. He notices the limiting factor on the plants is availability of light. Jim places the aquarium in a place where it will get more sunlight.
What will happen to the underwater plants?
Responses
Existing plants will grow larger and new plants will grow.Existing plants will grow larger and new plants will grow.
Existing plants will become smaller, but new plants will grow.Existing plants will become smaller, but new plants will grow.
Existing plants will become smaller and no new plants will grow.Existing plants will become smaller and no new plants will grow.
Existing plants will grow larger, but no new plants will grow.Existing plants will grow larger, but no new plants will grow.
question 7
A population of chipmunks that live in the woods near a town has learned to raid trash cans. The population of chipmunks is increasing rapidly.
The city replaces the existing trash cans with new, “chipmunk-proof” trash cans.
What most likely will happen to the population of chipmunks in the short term and long term?
Responses
The population will not change in the short term. In the long term, the population will increase dramatically.The population will not change in the short term. In the long term, the population will increase dramatically.
The population will not change in the short or long term. The chipmunks will switch to other food sources.The population will not change in the short or long term. The chipmunks will switch to other food sources.
The population will drop in the short term. In the long term, the population will stabilize as chipmunks switch to other food sources.The population will drop in the short term. In the long term, the population will stabilize as chipmunks switch to other food sources.
The chipmunks do not have other food sources. The populationwill drop in both the short and long term.
Step-by-step explanation: When a plant gets more sunlight it will grow very fast and get larger then it will grow some more
The measure of an interior angle of a regular polygon is 162º. Find the number of sides in the polygon.
_______sides
Answer:
20 sides
Step-by-step explanation:
The number of sides of a regular polygon, if each of its interior angles is 162° is 20.
please help fast and thank you
The order of the functions' average rates is h(x), g(x) and f(x)
How to order the functions' average ratesFrom the question, we have the following parameters that can be used in our computation:
The functions
The interval is given as
4 ≤ x ≤ 7
And the average rate is calculated as
Rate = (f(x₂) - f(x₁))/(x₂ - x₁)
So, we have
f(x) rate = (10 + 10)/(7 - 4) = 6.67
g(x) rate = (19 - 7)/(7 - 4) = 4
For h(x), we have
h(7) = 7^2 - 9 * 7 + 5 = -9
h(4) = 4^2 - 9 * 4 + 5 = -15
So, we have
h(x) rate = (-15 + 9)/(7 - 4) = -2
When compared, we have
-2 less than 4 less than 6.67
So, the order is h(x), g(x) and f(x)
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3. In a picture of a woman sweeping with a broom, her broom measures 2cm long. In reality, it is 1m long. In the same picture, a wall measures 5cm in height. In reality, how tall is the wall?
g suppose that w(t) is a real-valued signal. prove that its fourier transform magnitude is symmetric around f
Since the Fourier transform of a real value signal is the product of the even cosine transform and the odd sine transform, it is conjugate symmetric and consequently symmetric about the frequency 0.
A real-valued signal's Fourier transform is a conjugate symmetric function, which means it is symmetric about frequency 0. This is demonstrated by the fact that a real-valued signal's Fourier transform may be expressed as the product of its cosine and sine transforms:
F(w) = Fc(w) plus Fs (w)
Where the cosine and sine transforms, respectively, are denoted by Fc(w) and Fs(w). Both the sine and cosine transforms are symmetric about the frequency zero because the sine transform is an odd function and the cosine is an even function. As a result, a real-valued signal's Fourier transform also exhibits symmetry near frequency 0.
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The complete question is
g suppose that w(t) is a real-valued signal. prove that its fourier transform magnitude is symmetric around frequency 0.
Write a word problemthat could’ve solved by one of the problem
The required word problem of expression 2(5+w)=42, can be given as twice the sum of a number, and 5 is 42,
What are equation models?The equation model is defined as the model of the given situation in the form of an equation using variables and constants.
Here,
The question seems to be incomplete, from sources complete expression is given as 2(5+w)=42,
So word problem of expression 2(5+w)=42 can be given as,
Twice the sum of a number and 5 is 42, when we sum a number with 5 and doubled that number gives 42 as a product.
Thus, the required word problem of expression 2(5+w)=42, can be given as twice the sum of a number, and 5 is 42,
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What are m and b in the linear equation, using the common meanings of m and b?
2 + 3x + 5 - 2x = y
The values of m and b on the linear equation:
2 + 3x + 5 - 2x = y
are
m = 1
b = 7
How to find the values of m and b?A general linear equation can be written as in the slope-intercept form as:
y = m*x + b
Where m is the slope and b is the y-intercept.
Here we have the linear equation
2 + 3x + 5 - 2x = y
Now we want to write this in the slope-intercept form, to do so, we just need to simplify the left side, then we will get:
2 + 3x + 5 - 2x = y
(2 + 5) + (3x - 2x) =y
7 + x = y
Then the linear equation is:
y = 1*x + 7
So the values are:
a = 1
b = 7
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What is the measure of AC
Side AC of the similar triangle system has a length of 6 units. (Correct choice: C)
How to determine the missing length in a system of two similar right triangles
Two triangles are similar when both have congruent angles but sides are not congruent though proportional. In this problem we find the case of two similar right triangles, whose relationship is shown below:
AD / AC = CD / BC = AC / AB
If we know that AD = 4 and AB = 9, then the length of side AC is:
4 / AC = AC / 9
AC ² = 36
AC = 6
The length of side AC is equal to 6 units.
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Order the steps to solve the equation
log(x2-15) = log(2x) form 1 to 5.
x²-2x-15=0
Potential solutions are -3 and 5
x²-15 = 2x
x-5=0 orx+3=0
(x - 5)(x + 3) = 0
A logarithm is an operation used to determine the exponent of a given base number.
What is logarithm?
A logarithm is a mathematical function that describes the relationship between the power to which a number (called the base) must be raised, and the resulting value. The logarithm of a number is the power to which the base must be raised to produce that number.
Start by moving the logarithm to the left-hand side of the equation: log(x²-15) = log(2x)
Next, use the property of logarithms that log(a/b) = log(a) - log(b) to get x²-15 = 2x
then you can simplify the equation by subtracting 2x from both sides x²-2x-15=0
then factor the equation to get (x - 5)(x + 3) = 0
Finally, use the Zero Product Property to solve for x. The solutions will be x = 5 or x = -3
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The highest elevation in California is 14,505 feet, and the lowest elevation is −282 feet. Write and then evaluate an expression to determine the distance between the lowest and highest elevations.
The expression to determine the distance between the lowest and the highest point is -
{x} = 14505 + |- 282|
What are algebraic expressions?In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax.Given is that the highest elevation in California is 14,505 feet, and the lowest elevation is − 282 feet.
The expression to determine the distance between the lowest and the highest point can be written as -
{x} = 14505 + |- 282|
where -
| | represents modulus function.
Therefore, the expression to determine the distance between the lowest and the highest point is -
{x} = 14505 + |- 282|
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for each of the indefinite integrals below, choose which of the following substitutions would be most helpful in evaluating the integral. enter the appropriate letter (a,b, or c) in each blank. do not evaluate the integrals. A. x= 6 tan θ B. x= 6 sin θ C. x= 6 sec θ 1. Integral (x^2 - 36)^5/2 dx 2. Integral (x^2 dx/ (root 36+x^2) 3. Integral (dx/ (root 36+x^2)^3 4. Integral (root x^2+36) dx 5. Integral x^2 (root 36+x^2)
We can use the integral x=4secθ for each of the indefinite integrals below.
In the given question we can use either 4tanθ,4sinθ or 4secθ to substitute and simplify the given indefinite integrals,
the 1st integral is,
∫ √(x²-16) dx
we can use the trigonometric substitution x=4secθ in this integral, due to which after simplification it will become as,
∫16 secθtan²θ dθ
which then can easily be solved.
the 2nd integral is,
∫ x²/√(x²-16) dx
we can use the trigonometric substitution x=4sinθ in this integral, due to which after simplification it will become as,
∫16 sin²θ dθ
which then can be solved by using a reduction formula.
the 3rd integral is,
∫ dx /(16+x²)³
we can use the trigonometric substitution x=4tanθ in this integral, due to which after simplification it will become as,
∫(1/16⁴) cos⁴θ dθ
the 4th integral is,
∫ (x²-16)^(5/2)dx
we can use the trigonometric substitution x=4secθ in this integral, due to which after simplification it will become as,
∫4096 secθtan⁶θ dθ
the 5th integral is,
∫ dx /(16-x²)^(3/2)
we can use the trigonometric substitution x=4sinθ in this integral, due to which after simplification it will become as,
∫(1/16² cos²θ) dθ
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