Answers
5. Let [tex]x = \sin(\theta)[/tex]. Note that we want this variable change to be reversible, so we tacitly assume 0 ≤ θ ≤ π/2. Then
[tex]\cos(\theta) = \sqrt{1 - \sin^2(\theta)} = \sqrt{1 - x^2}[/tex]
and [tex]dx = \cos(\theta) \, d\theta[/tex]. So the integral transforms to
[tex]\displaystyle \int \frac{x^3}{\sqrt{1-x^2}} \, dx = \int \frac{\sin^3(\theta)}{\cos(\theta)} \cos(\theta) \, d\theta = \int \sin^3(\theta) \, d\theta[/tex]
Reduce the power by writing
[tex]\sin^3(\theta) = \sin(\theta) \sin^2(\theta) = \sin(\theta) (1 - \cos^2(\theta))[/tex]
Now let [tex]y = \cos(\theta)[/tex], so that [tex]dy = -\sin(\theta) \, d\theta[/tex]. Then
[tex]\displaystyle \int \sin(\theta) (1-\cos^2(\theta)) \, d\theta = - \int (1-y^2) \, dy = -y + \frac13 y^3 + C[/tex]
Replace the variable to get the antiderivative back in terms of x and we have
[tex]\displaystyle \int \frac{x^3}{\sqrt{1-x^2}} \, dx = -\cos(\theta) + \frac13 \cos^3(\theta) + C[/tex]
[tex]\displaystyle \int \frac{x^3}{\sqrt{1-x^2}} \, dx = -\sqrt{1-x^2} + \frac13 \left(\sqrt{1-x^2}\right)^3 + C[/tex]
[tex]\displaystyle \int \frac{x^3}{\sqrt{1-x^2}} \, dx = -\frac13 \sqrt{1-x^2} \left(3 - \left(\sqrt{1-x^2}\right)^2\right) + C[/tex]
[tex]\displaystyle \int \frac{x^3}{\sqrt{1-x^2}} \, dx = \boxed{-\frac13 \sqrt{1-x^2} (2+x^2) + C}[/tex]
6. Let [tex]x = 3\tan(\theta)[/tex] and [tex]dx=3\sec^2(\theta)\,d\theta[/tex]. It follows that
[tex]\cos(\theta) = \dfrac1{\sec(\theta)} = \dfrac1{\sqrt{1+\tan^2(\theta)}} = \dfrac3{\sqrt{9+x^2}}[/tex]
since, like in the previous integral, under this reversible variable change we assume -π/2 < θ < π/2. Over this interval, sec(θ) is positive.
Now,
[tex]\displaystyle \int \frac{x^3}{\sqrt{9+x^2}} \, dx = \int \frac{27\tan^3(\theta)}{\sqrt{9+9\tan^2(\theta)}} 3\sec^2(\theta) \, d\theta = 27 \int \frac{\tan^3(\theta) \sec^2(\theta)}{\sqrt{1+\tan^2(\theta)}} \, d\theta[/tex]
The denominator reduces to
[tex]\sqrt{1+\tan^2(\theta)} = \sqrt{\sec^2(\theta)} = |\sec(\theta)| = \sec(\theta)[/tex]
and so
[tex]\displaystyle 27 \int \tan^3(\theta) \sec(\theta) \, d\theta = 27 \int \frac{\sin^3(\theta)}{\cos^4(\theta)} \, d\theta[/tex]
Rewrite sin³(θ) just like before,
[tex]\displaystyle 27 \int \frac{\sin(\theta) (1-\cos^2(\theta))}{\cos^4(\theta)} \, d\theta[/tex]
and substitute [tex]y=\cos(\theta)[/tex] again to get
[tex]\displaystyle -27 \int \frac{1-y^2}{y^4} \, dy = 27 \int \left(\frac1{y^2} - \frac1{y^4}\right) \, dy = 27 \left(\frac1{3y^3} - \frac1y\right) + C[/tex]
Put everything back in terms of x :
[tex]\displaystyle \int \frac{x^3}{\sqrt{9+x^2}} \, dx = 9 \left(\frac1{\cos^3(\theta)} - \frac3{\cos(\theta)}\right) + C[/tex]
[tex]\displaystyle \int \frac{x^3}{\sqrt{9+x^2}} \, dx = 9 \left(\frac{\left(\sqrt{9+x^2}\right)^3}{27} - \sqrt{9+x^2}\right) + C[/tex]
[tex]\displaystyle \int \frac{x^3}{\sqrt{9+x^2}} \, dx = \boxed{\frac13 \sqrt{9+x^2} (x^2 - 18) + C}[/tex]
2(b). For some constants a, b, c, and d, we have
[tex]\dfrac1{x^2+x^4} = \dfrac1{x^2(1+x^2)} = \boxed{\dfrac ax + \dfrac b{x^2} + \dfrac{cx+d}{x^2+1}}[/tex]
3(a). For some constants a, b, and c,
[tex]\dfrac{x^2+4}{x^3-3x^2+2x} = \dfrac{x^2+4}{x(x-1)(x-2)} = \boxed{\dfrac ax + \dfrac b{x-1} + \dfrac c{x-2}}[/tex]
5(a). For some constants a-f,
[tex]\dfrac{x^5+1}{(x^2-x)(x^4+2x^2+1)} = \dfrac{x^5+1}{x(x-1)(x+1)(x^2+1)^2} \\\\ = \dfrac{x^4 - x^3 + x^2 - x + 1}{x(x-1)(x^2+1)^2} = \boxed{\dfrac ax + \dfrac b{x-1} + \dfrac{cx+d}{x^2+1} + \dfrac{ex+f}{(x^2+1)^2}}[/tex]
where we use the sum-of-5th-powers identity,
[tex]a^5 + b^5 = (a+b) (a^4-a^3b+a^2b^2-ab^3+b^4)[/tex]
Related Questions
(-24uy^2+19u^3y^5)/(-3u^3y^5)
Simplify
Answers
Answer:
8/(u^2y^3) -19/3
Step-by-step explanation:
The relevant rules of exponents are ...
(a^b)/(a^c) = 1/a^(c-b)
a^0 = 1
__
Your expression can be simplified like this:
[tex]\dfrac{-24uy^2+19u^3y^5}{-3u^3y^5}=\dfrac{24uy^2}{3u^3y^5}-\dfrac{19u^3y^5}{3u^3y^5}=\boxed{\dfrac{8}{u^2y^3}-\dfrac{19}{3}}[/tex]
Simplify the expression. –(3)–4
a)12
b)[tex]-\frac{1}{81}[/tex]
c)81
d)[tex]\frac{1}{81}[/tex]
Answers
Answer:
Answer is {C}
Step-by-step explanation:
Which will usually always be a higher amount?
A. Compound interest
B. Time
C. Debit
D. Simple interest
Answers
Solve for x: 3 (square root2-2x)+1=13
Answers
Answer:
x=7
Step-by-step explanation:
3√(2-2x) + 1 = 13
subtract 1 from both sides
3√(2-2x) = 12
Divide 3 from both sides
√2-2x= 4
square both sides
2-2x = 16
subtract 2 from both sides
2x= 14
x=7
You spin the spinner once.
234567
What is P(6)?
Write your answer as a fraction
or whole number.
Helps pls i did 120 questions
Answers
Answer:
1/6
Step-by-step explanation:
There are 6 possible numbers for the spinner to land on, and only 1 of those cases will be where the spinner lands on 6.
Therefore, the answer is 1/6
5>x/5
Show all work in steps
Answers
Answer:
[tex]\boxed{\sf{x < 25}}[/tex]Step-by-step explanation:
To find the value of x, isolate it on one side of the equation.
5>x/5First, you have to switch sides.
→ x/5<5
Then, you multiply by 5 from both sides.
→ 5x/5<5*5
Solve.
Multiply the numbers from left to right.
→ 5*5=25
→ x<25
Therefore, the solution is x<25, which is our answer.I hope this helps you! Let me know if my answer is wrong or not.
Answer:
25 > x
Step-by-step explanation:
[tex]5 > \frac{x}{5}[/tex]
To find the value of x, you have to multiply both sides by 5.
[tex]5*5 > \frac{x}{5}* 5[/tex]
[tex]25 > \frac{5x}{5}[/tex]
[tex]25 > x[/tex]
$600 invested at 6% compounded annually for six years
Answers
Answer:
The final balance is $851.11.
The total compound interest is $251.11.
Step-by-step explanation:
Q10. Consider the following algorithm:
g_1 = 9
g_2 = 5
Git
For k starting at 0 and ending with 8:
gk = (k-1)*gk-1 + gk-2
What is the last term, 8, of the recursive sequence generated as a result of
executing this algorithm?
Answers
Answer:
61,940
Step-by-step explanation:
For a recursive sequence of reasonable length, it is convenient to use a suitable calculator for figuring the terms of it. Since each term not only depends on previous terms, but also depends on the term number, it works well to use a spreadsheet for doing the calculations. The formula is easily entered and replicated for as many terms as may be required.
__
The result of executing the given algorithm is shown in the attachment. (We have assumed that g_1 means g[-1], and that g_2 means g[-2]. These are the starting values required to compute g[0] when k=0.
That calculation looks like ...
g[0] = (0 -1)×g[-1] +g[-2} = (-1)(9) +5 = -4
The attachment shows the last term (for k=8) is 61,940.
I am a polygon that has two pairs of opposite sides that are parallel what am I
Answers
A Parallelogram because it is a quadrilateral with 2 pairs of parallel sides!
y = x^2 + 8x + 10
Vertex:
Axis of Symmetry:
Y-intercept:
Answers
Vertex: (-4, -6)
Axis of symmetry: x= -4
Y-intercept: y=10
Have a great day
The vertex of the function is (-4, -6), the axis of symmetry is x= -4 and the y-intercept is (0, 10).
What is the function?Functions are the fundamental part of the calculus in mathematics. The functions are the special types of relations. A function in math is visualized as a rule, which gives a unique output for every input x.
The given function is y=x²+8x+10.
Rewrite in vertex form and use this form to find the vertex (h, k).
(-4, -6)
The axis of symmetry is x= -4
To find the x-intercept, substitute in 0 for y and solve for x. To find the y-intercept, substitute in 0 for x and solve for y.
x-intercept(s): (-4+√6, 0), (-4-√6, 0)
y-intercept(s): (0, 10)
Therefore, the y-intercept is (0, 10).
To learn more about the function visit:
https://brainly.com/question/28303908.
#SPJ7
1 Whatis the value ofthe expression y 1/2 +z)+6 ( 2 + when y= 2 and 7 = 3 ? 4
.A) 57/8
.B) 19/2
.C) 17/2
.D) 22/3
30 BRAINLY points plus user!!!
Answers
2(1/2+3/4)+6=
17/2 or 8.5
The answer is C
Answer:
Option C
Step-by-step explanation:
Given:
[tex]y = 2[/tex]
[tex]z = \dfrac{3}{4}[/tex]
Substitute the values into the expression:
⇒ y(1/2 + z) + 6
⇒ 2(1/2 + 3/4) + 6
Make common denominators in the parenthesis:
⇒ 2(1/2 + 3/4) + 6
⇒ 2(2/4 + 3/4) + 6
Simplifying the expression:
⇒ 2(5/4) + 6
⇒ 2 × 5/4 + 6
**Keep in mind that multiplication first, then addtion
⇒ 10/4 + 6
Make common denominators:
⇒ 10/4 + 24/4
⇒ 34/4
Simplify the fraction:
⇒ 17/2 (Option C)
20 POINTS PLS
How to tell if your data has a wide range or a small range?
Answers
Directions: Type the correct answer in each box. Spell all words correctly, and use numerals instead of words for numb
For this problem any non-integer answers should be entered as fractions in simplest form using / as the fraction bar(s).
A spinner has five colored sections of different sizes. Felicia spins the arrow on the spinner and records the color of the section it
lands on in the table below.
Complete the statements based on the given table.
The experimental probability of landing on orange is__
The experimental probability of landing on___is 1/5
The experimental probability of landing on a color that is not blue is___
Answers
Answer:
1/3, Yellow, 3/4
Step-by-step explanation:
All the numbers add up to 60, so using fractions
Red is 11/60
Green is 2/60
Yellow is 12/60
Blue is 15/60
Orange is 20/60
Finding The probability of w/o blue is subtracting 15 from 60 which ends as 45/60, and simplifying that would be 3/4
Find the area of the triangle.
16 ft
62 ft
Not drawn to scale.
A. 496 ft2
B. 248 ft?
C. 78 ft2
D. 37 ft2
Answers
Answer:
A. 496 ft2
Step-by-step explanation:
area of a triangle is
1/2 times base times height
1/2 times 16 times 62 =
496
I will give Brainliest! :<
This meal costs $75.00.
Fatima used a 20% off coupon and then tax was added.
The total after the coupon and tax was $75.00.
What was the tax rate?
Submit and EXPLAIN, please!
Answers
Answer:
25 %
Step-by-step explanation:
75 - 20% = 60
60 + 60 (x) = 75 where x = tax rate
x = .25 or 25 %
Ik this cause I took the test
1) Which fractions are equivalent to 6/12? Choose ALL the correct answers.
12
Answers
Answer:
Infinite
Step-by-step explanation:
Most common examples are 1/2 3/6 7/14 4/8 2/4 5/10 8/16 50/100
Simplify the polynomial then evaluate for x=-1 3x^2-4+2x-1+x^2
Answers
Answer:
-3
Step-by-step explanation:
Simplified polynomial: 4x^2 + 2x - 5
Plug in -1: 4(-1)^2 + 2(-1) - 5
= -3
Write (b^3)^2 without exponents
Answers
Answer:
(b)(b)(b)(b)(b)(b)
Step-by-step explanation:
[tex](b^3)^2=b^6[/tex]
(b)(b)(b)(b)(b)(b)
Answer:
Step-by-step explanation:
(b^3)^2
= b^(3*2)
= b^6
= b*b*b*b*b*b.
PLEASE HELP!!!!! I"LL MARK AS BRAINLIEST!!!Tom conducted a drawing competition at school. He bought 5 boxes of colored pencils and spent $5.25 on drawing paper. However, more contestants entered than he expected, and he had to buy 2 more boxes of colored pencils. The total cost of the paper and pencils was $82.95. Identify the equation and solution that would help to find the cost of each box of pencils.
A. 5x + 2x + 5.25 = 11.10; $82.95
B. 5x − 2x + 5.25 = 82.95; $11.10
C. 5x − 2x + 5.25 = 82.95; $1.11
D. 5x + 2x + 5.25 = 82.95; $11.10
Answers
A student performed the following steps to find the solution to the equation x^2 + 14x + 40 = 0. Where did the student go wrong?
Step 1. Factor the polynomial into (x + 4) and (x + 10)
Step 2. x + 4 = 0 or x + 10 = 0
Step 3. x = -4 or x = 10
A. In step 3
B. The student did not make any mistakes, the solution is correct
C. In step 1
D. In step 2
Answers
Answer:
A. in Step 3
Step-by-step explanation:
It is factored correctly. Then separated into two smaller equations correctly. But in solving x + 10 = 0, there is an error.
x + 10 = 0
Add -10 to both sides.
x = -10
So, the error is in step 3.
Answer:
A . In Step 3.
Step-by-step explanation:
In step 3 the student went wrong
x = 4 or x = -10 is correct
( if x + 10 = 0, then x = -10)
HELP I GOT 2 mins please
Answers
Answer:
x=27 degrees
EAF= 27
Step-by-step explanation:
sorry if this isnt right i tried my best
Find the area and the circumference of a circle with radius 9 ft.
Use the value 3.14 for pie, and do not round your answers. Be sure to include the correct units in your answers.
Answers
Answers:
Area = 254.34 square feet
Circumference = 56.52 feet
=========================================================
Work Shown:
Area = pi*(radius)^2
A = pi*r^2
A = 3.14*9^2
A = 254.34 square feet
----------------
Circumference = 2*pi*(radius)
C = 2*pi*r
C = 2*3.14*9
C = 56.52 feet
The two results are approximate since pi = 3.14 is approximate.
1st area of circle:
[tex]\bold{\ area \ of \ circle \ : \pi r^2}[/tex]
[tex]\hookrightarrow \sf (3.14)(9)^2[/tex]
[tex]\hookrightarrow \sf 254.34 \ ft^2[/tex]
2nd circumference of circle:
[tex]\bold{circumference \ of \ circle \ : \ 2\pi r}[/tex]
[tex]\hookrightarrow \sf 2(3.14)(9)[/tex]
[tex]\hookrightarrow \sf 56.52 \ ft[/tex]
area of circle = 254.34 ft²circumference of a circle = 56.52 ftHELP QUICK PLEASE!!!
Answers
Answer:
19
please make this answer brainliest If I helped you!
Answer:
c=19
Step-by-step explanation:
110-91=19
129-110=19
148-129=19
4x19 = 76 (91-76=15)
76+15 =91
5x19= 95 ( 110-95=15)
95+15= 110
6x19 = 114 (129-114 =15)
114+15=129
7x19=133 ( 148-133 =15)
133+15 = 148
Part B
Find the equation of a line passing through the same point that f(x) passes through at x = 3 with a slope equal to the
limit you found in part A.
Answers
The function is f(x) = x³ − 4x² + 2.
How to find the slope?The slope of a line or straight object is the ratio of how much amount of rise occurs in correspondence to the increment in the run.
Find the difference between the y coordinates, Δy is change in y.Find the difference between the x coordinates, Δx is change in x.Divide Δy by Δx to find slope.As,
Slope = rise/ run
Given:.
f(x) = x³ − 4x² + 2
As a line passing through the same point that f(x) passes through at x = 3 with a slope equal to the limit.
Then the slope of the function is given by the differentiation.
slope= d/dx [f(x)]
slope= d/dx ( x³ − 4x² + 2)
slope= 3x² - 8x
slope= 3* 3* 3 -8*3
slope= 27-24
slope= 3
Learn more about the slope here:
brainly.com/question/2503591
#SPJ2
Business mathematics.
Mr Kula opens a shopfront where he will lend money to anyone ( without security) if they are willing to pay his interest rate of 2% per week compounded daily. The first customer borrows $5000 to buy a used car, but is unable to pay Mr Kula back for 12 weeks. How much will the customer have to pay back at this time?
Answers
Answer:
$6200
Step-by-step explanation:
2% of $5000 = $100
So that's a hundred dollars per week
12 weeks x $100 = $1200
$5000 (The original amount the first customer borrowed)
+ $1200 (The interest rate)
= $6200
(P.S. Please give me a brainliest if you can)
Complete part (a) and part (b) given the following system of equations x-y=0 2x + 3y = -5 Solve this system graphically.
Answers
Answer:
(-1, -1)
Step-by-step explanation:
x - y = 0 ------> x = y
2x + 3y = -5
2(y) + 3y = -5
5y = -5
y = -1
x - (-1) = 0
x = -1
The solution to this system is (-1, -1). To solve graphically, plot both functions and you will see that they intersect at (-1, -1).
I hope this helps!
please help me Find m< G
Answers
Answer:
70°
Step-by-step explanation:
All angles in a kite equal 360°.∠F≅∠H. 360-(90+90+110) will get you the answer.
Three boxes are stacked one on top of the other. One box is 6 feet 6 inches tall, one is 4 feet 11 inches tall, and one is 5 feet 8 inches tall. How high is the stack? Write your answer in feet and inches. Use a number less than 12 for inches.
Answers
Heyo xD! Its easy since we have been asked to find out the total height of the stack which will be equal to the sum of height of each box. Thus, we can easily add them up and get our answer.
Given :-
Box 1 = 6 ft. 6 inches Box 2 = 4 ft. 11 inchesBox 3 = 5 ft. 8 inchesTo Find :-
The height of the stackSolution :-
Height of the stack,
=> 6 ft. 6 inches + 4 ft. 11 inches + 5 ft. 8 inches
=> 15 ft. 15 inches
And we know that 12 inches = 1 ft.
Thus, 15 inches = 1 ft. 3 inches
Thus,
=> 15ft. + 1 ft. 3 inches
=> 16 ft. 3 inches
Thus, the height of the stack is 16 ft. 3 inches.
Answer:
17 feet and 1 inch
Step-by-step explanation:
6 feet and 6 inches + 4 feet and 11 inches equals 10 feet and 17 inches.
10 feet and 17 inches + 5 feet and 8 inches equals 15 feet and 25 inches.
When converted the answer is 17 feet and 1 inch. This is so because 25 divided by 12 equals 2 remainder 1 . 2 feet plus 15 feet equals 17 feet and the remainder is 1 which is 1 inch.
So, the answer is 17 feet and 1 inch.
Cole is buying a pair of sneakers. The expression shown represents the sales tax on the price of the sneakers, p.
0.045p
Write an expression in terms of p to represent the total amount Cole spends on the pair of sneakers, including tax.
Answers
By direct addition, we will see that the expression is:
T = 1.045*P.
What expression represents the total price of the sneakers?
We know that the price of the sneakers is P. We also know that the expression which represents the taxes of the sneakers is 0.045*P.
Now, we can assume that the total price of the sneakers will be the price plus the correspondent tax, then we can write:
T = P + 0.045*P = (1 + 0.045)*P = 1.045*P
This is the expression that represents the total amount Cole spends on the pair of sneakers.
If you want to learn more about addition, you can read:
https://brainly.com/question/11562149
Find the value of x.
52°
128°
38°
xº
x = [21°
Answers
Step-by-step explanation:
x = 360 - 128 + 52 + 38
x = 360 - 218
x = 142