Answer:
[tex]\int\limits {\frac{x}{x^4 + 16}} \, dx = \frac{1}{8}*arctan(\frac{x^2}{4}) + c[/tex]
Step-by-step explanation:
Given
[tex]\int\limits {\frac{x}{x^4 + 16}} \, dx[/tex]
Required
Solve
Let
[tex]u = \frac{x^2}{4}[/tex]
Differentiate
[tex]du = 2 * \frac{x^{2-1}}{4}\ dx[/tex]
[tex]du = 2 * \frac{x}{4}\ dx[/tex]
[tex]du = \frac{x}{2}\ dx[/tex]
Make dx the subject
[tex]dx = \frac{2}{x}\ du[/tex]
The given integral becomes:
[tex]\int\limits {\frac{x}{x^4 + 16}} \, dx = \int\limits {\frac{x}{x^4 + 16}} \, * \frac{2}{x}\ du[/tex]
[tex]\int\limits {\frac{x}{x^4 + 16}} \, dx = \int\limits {\frac{1}{x^4 + 16}} \, * \frac{2}{1}\ du[/tex]
[tex]\int\limits {\frac{x}{x^4 + 16}} \, dx = \int\limits {\frac{2}{x^4 + 16}} \,\ du[/tex]
Recall that: [tex]u = \frac{x^2}{4}[/tex]
Make [tex]x^2[/tex] the subject
[tex]x^2= 4u[/tex]
Square both sides
[tex]x^4= (4u)^2[/tex]
[tex]x^4= 16u^2[/tex]
Substitute [tex]16u^2[/tex] for [tex]x^4[/tex] in [tex]\int\limits {\frac{x}{x^4 + 16}} \, dx = \int\limits {\frac{2}{x^4 + 16}} \,\ du[/tex]
[tex]\int\limits {\frac{x}{x^4 + 16}} \, dx = \int\limits {\frac{2}{16u^2 + 16}} \,\ du[/tex]
Simplify
[tex]\int\limits {\frac{x}{x^4 + 16}} \, dx = \int\limits {\frac{2}{16}* \frac{1}{8u^2 + 8}} \,\ du[/tex]
[tex]\int\limits {\frac{x}{x^4 + 16}} \, dx = \frac{2}{16}\int\limits {\frac{1}{u^2 + 1}} \,\ du[/tex]
[tex]\int\limits {\frac{x}{x^4 + 16}} \, dx = \frac{1}{8}\int\limits {\frac{1}{u^2 + 1}} \,\ du[/tex]
In standard integration
[tex]\int\limits {\frac{1}{u^2 + 1}} \,\ du = arctan(u)[/tex]
So, the expression becomes:
[tex]\int\limits {\frac{x}{x^4 + 16}} \, dx = \frac{1}{8}\int\limits {\frac{1}{u^2 + 1}} \,\ du[/tex]
[tex]\int\limits {\frac{x}{x^4 + 16}} \, dx = \frac{1}{8}*arctan(u)[/tex]
Recall that: [tex]u = \frac{x^2}{4}[/tex]
[tex]\int\limits {\frac{x}{x^4 + 16}} \, dx = \frac{1}{8}*arctan(\frac{x^2}{4}) + c[/tex]
Pls help me this is 7th grade (ADVANCED)
Answer:
Step-by-step explanation:
Any line will add up to 180 degrees. If you subtract the 165 degrees from 180 (total), you will get the x value.
180 - 165 = 15 degrees
What is the solution to this equation?
x + 14 = -6
Answer:
x = -20
Step-by-step explanation:
Find the surface area of this rectangular prism. Be sure to include the correct unit in your answer.
7m
6 m
2 m
The length of a rectangle is three times the width. The perimeter of the rectangle is 32 inches. What is the area of the rectangle (in square inches)?
Answer:
48 square inches
plz mark brainliest
Step-by-step explanation:
l= 3w
perimeter = 2(l+w) =2(3w+w) = 2(4w)= 8w
32=8w
w=32/8 =4 inches
l= 3(4) =12 inches
area = l×w = 12×4 = 48 square inches
Which expression is equivalent to 6(3c-4)-2(5c-1)
Answer:
Step-by-step explanation:
8c−22
Answer:
8c−22
Step-by-step explanation:
WILL MARK BRAINLYIST use the figure to determine each angle measure
Answer:
1. 60 2. 65 3. 115 4. 55 5. 120 6. 60
Step-by-step explanation:
Hope it helps!
police chase a car. police are originally 5 miles behind. if the police car's speed is 90 mph and the car they chase is moving with a speed of 80 mph. how long will it take them to catch up?
Step-by-step explanation:
Let x be the number of hours for
the police to catch up to the car.
We have 90x = 80x + 5.
=> 10x = 5
=> x = 0.5.
Hence it will take them 30 minutes.
Evaluate 2 to the ninth power. Help plz
2^9 = 2 x 2 x2 x 2 x 2 x 2 x2 x 2 x 2 = 512
2 ^9 = 512
Exit Ticket 1/12 (A.2E, A.ZF)
What is the slope of a line perpendicular to a line with slope -6/5?
Answer:
5/6
Step-by-step explanation:
Perpendicular lines have slopes that are negative reciprocals of each other.
-6/5
6/5
5/6,,
PLEASE HELP ME QUICKLY PLEASE SOMEBODY I WILL GIVE BRAINLIEST
Answer:
.
Step-by-step explanation:
Find the lcd between 1/6 and 2/9. Show your work.
Which expression can be used to determine how far around a rider has traveled after the Ferris wheel has gone around 8 times?
ANSWER FAST PLEASE
Answer:
Its A
Step-by-step explanation:
But don't believe me this is wrong
HELP AGAIN PLEASE 20 POINTS What methods can be used to solve a system of equations and when would you use each? Be sure to include systems with nonlinear equations.
Answer:
The three ways to solve a system of equations are substitution, elimination, and graphing. You can use these methods to solve any system of equations, it just depends on which method you use.
Step-by-step explanation:
In the substitution method, you get one variable isolated on one side of an equation, then use that equation to substitute the variable with what it's equal to into the second equation to solve the equation.
In the elimination method, you get all the variables in your equations together one side, and the constant on the other. Then, multiply or divide one or two equations to get one variable with the same coefficient for both equations (be sure to multiply or divide both sides to each equation), and add or subtract the equations together to get rid of that variable (make sure to add the left sides of the equations together and the right sides of the equations together, no the opposite!) and solve the equation.
In the graphing method, you solve for y in each equation, graph both equations on a coordinate plane, and find the point of intersection for both lines to solve the equation (For example, if the point of intersection is (2, 7), the answer to your equation would be X = 2 and Y = 7).
b. , using the
Pythagorean Theorem,
show that.
C. Solve for X
Define/explain the Pythagorean theorem.
Answer: The Pythagorean theorem consists of a formula a^2+b^2=c^2 which is used to figure out the value of (mostly) the hypotenuse in a right triangle. It's a fundamental relation in Euclidean geometry among the three sides of a right triangle.
Step-by-step explanation:
Answer:
Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.
What is the decimal equivalent for 139%
Answer: 1.39
Step-by-step explanation:
Hope it helps! Mark me brainliest.
Mhanifa please help!! I will mark brainliest. Random answers will be reported
Answer:
[tex]\sqrt{h^2 + k^2}[/tex]Step-by-step explanation:
Point O has coordinates (0, 0) B has coordinates (h, k)C has coordinates (h, 0)Use the distance formula to find the length of OB:
OB = [tex]\sqrt{(h-0)^2 + (k-0)^2}[/tex] = [tex]\sqrt{h^2 + k^2}[/tex]у= 4х + 3
у = -X - 2
Answer:
x = - 1
Step-by-step explanation:
y = 4x + 3
y = -x - 2
Find x:
4x + 3 = -x - 2
5x + 3 = - 2 (Adding x to both sides)
5x = - 5 (Subtracting 3 from both sides)
x = -1 (Dividing 5 from both sides)
Answer: x = - 1
4. MARBLES Virginia's
mother gave her marbles for
her birthday. Virginia lost 13
of them. If she has 24 marbles
left, write and solve an
equation to find how many her
mother gave her.
Answer:
37
Step-by-step explanation:
If she has 24 left, and lost 13, you add 13 and 24 to find the original amount.
Virginia's mother gave her 37 marbles for her birthday.
What is an expression?Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
We have to given that;
Virginia's mother gave her marbles for her birthday.
Let number of marbles = x
Here, Virginia lost 13 of them and she has 24 marbles left.
So, We get;
⇒ x - 13 = 24
Solve for x;
⇒ x = 24 + 13
⇒ x = 37
Thus, Virginia's mother gave her 37 marbles for her birthday.
Learn more about the mathematical expression visit:
brainly.com/question/1859113
#SPJ2
Pls help Find the measure of angle B
Answer:
∠b = 61°
Step-by-step explanation:
∠b + 29° = 90°
∠b = 90° - 29°
∠b = 61°
A set of equations is given below:
Equation A: y = x + 9
Equation B: y = 2x + 1
What is the solution (
,
) to the given set of equations?
Answer:
comparing both equation we get,
2X+1=X+9
or, X=8
Y=X+9
or, Y=17
therefore the required solution is (8,17)
nesesito ayuda con este problema
Answer:
Step-by-step explanation:
I KNOW SPEAK SPANISH GO TO A MEXICAN SITE NOT A AMERICAN SITE
find the solution
[tex]3(2x + 12) = 6(x + 6)[/tex]
The area of the parallelogram is 315 square units. What is the height of the parallelogram?
Answer:
h = 15 units
Step-by-step explanation:
A = bh
315 = 21h
h = 315/21
h = 15
Answer:
15 sq. units
Step-by-step explanation:
The Intuition for why the area of a parallelogram is A=bh (area = base x height)
The formula for the area of a parallelogram is base times height, just like the formula for the area of a rectangle.
Have a great day!
How do I do this? Please only the best help.
Answer:
77.5 degrees and 102.5 degrees
Step-by-step explanation:
Given that : one angle is25 degrees greater than another. Let the two angles be represented by a and b. Since one angle is 25 degreess greater than another, then:
b = a + 25 degrees
But;
a + b = 180 degrees (sum of angles on a straight line)
a + a + 25 degrees = 180 degrees
2a = 180 degrees - 25 degrees
a = 77.5 degrees
b = 77.5 degrees + 25 degrees = 102.5 degrees
Therefore the measures of all angles formed by the lines is either 77.5 degrees or 102.5 degrees.
Solve the system using the substitution method
x = -6y - 4
6x - 12y = 24
Answer:
x = 2, y = -1
Step-by-step explanation:
We have a solution system, x = -6y - 4 and 6x - 12y = 24.
Now, since are not only asked to use the substitution method, but at the same time know the value of x, let's plug the first system into the second.
x = -6y - 4
6(-6y - 4) - 12y = 24
-36y - 24 - 12y = 24
-48y - 24 = 24
Add 24 to both sides :
-48y = 48
Divide -48 by both sides :
y = -1
Now that we have y, we can replace it into the second system again to get x.
6x - 12(-1) = 24
6x + 12 = 24
Solve for x by first getting rid of 12 from both sides :
6x = 12
Divide 6 by both sides :
x = 2
Therefore, x = 2 and y = -1
PLEASE ANSWER I WILL MARK YOU AS BRAINLIEST IF YOU ANSWER CORRECTLYY PLS AND TY
Answer:
it is a function
Step-by-step explanation:
Domain is 7 and the range is all the x values
Please answer all of them
If 3 hippos eat 5 pumpkins, how many pumpkins per hippo is that?0.6 pumpkin per hippo
If there are 9 hippos, how many pumpkins do you need?
If 3 hippos eat 5 pumpkins, how many hippos per pumpkin is that?
If you have 30 pumpkins, how many hippos can you feed?
HELP WITH MY MATH PLEASE
Answer:
because im god and god is good and 9 x 10 is 99999
Step-by-step explanation:
Questions:
1. If the run of a ramp increases and the rise stays the same, how does the slope of the ramp change?
9514 1404 393
Answer:
slope decreases
Step-by-step explanation:
The slope of a ramp is directly proportional to the rise and inversely proportional to the run. When the run increases, the slope decreases.
