1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
exis [7]
3 years ago
7

Please, I need help in this ??

Mathematics
1 answer:
nignag [31]3 years ago
4 0

Answer:

\int\frac{x^{4}}{x^{4} -1}dx = x + \frac{1}{4} ln(x-1) - \frac{1}{4} ln(x+1)-\frac{1}{2} arctanx + c

Step-by-step explanation:

\int\frac{x^{4}}{x^{4} -1}dx

Adding and Subtracting 1 to the Numerator

\int\frac{x^{4} - 1 + 1}{x^{4} -1}dx

Dividing Numerator seperately by x^{4} - 1

\int 1 + \frac{1}{x^{4}-1 }\, dx

Here integral of 1 is x +c1 (where c1 is constant of integration

x + c1 + \int\frac{1}{(x-1)(x+1)(x^{2}+1)}\, dx----------------------------------(1)

We apply method of partial fractions to perform the integral

\frac{1}{(x-1)(x+1)(x^{2}+1)} = \frac{A}{x-1} + \frac{B}{x+1} + \frac{C}{x^{2} + 1}------------------------------------------(2)

\frac{1}{(x-1)(x+1)(x^{2}+1)} = \frac{A(x+1)(x^{2} +1) + B(x-1)(x^{2} +1) + C(x-1)(x+1)}{(x-1)(x+1)(x^{2} +1)}

1 = A(x+1)(x^{2} +1) + B(x-1)(x^{2} +1) + C(x-1)(x+1)-------------------------(3)

Substitute x= 1 , -1 , i in equation (3)

1 = A(1+1)(1+1)

A = \frac{1}{4}

1 = B(-1-1)(1+1)

B = -\frac{1}{4}

1 = C(i-1)(i+1)

C = -\frac{1}{2}

Substituting A, B, C in equation (2)

\int\frac{x^{4}}{x^{4} -1}dx = \int\frac{1}{4(x-1)} - \frac{1}{4(x+1)} -\frac{1}{2(x^{2}+1) }

On integration

Here \int \frac{1}{x}dx = lnx and \int\frac{1}{x^{2}+1 } dx = arctanx

\int\frac{x^{4}}{x^{4} -1}dx = \frac{1}{4} ln(x-1) - \frac{1}{4} ln(x+1) - \frac{1}{2} arctanx + c2---------------------------------------(4)

Substitute equation (4) back in equation (1) we get

x + c1 + \frac{1}{4} ln(x-1) - \frac{1}{4} ln(x+1) - \frac{1}{2} arctanx + c2

Here c1 + c2 can be added to another and written as c

Therefore,

\int\frac{x^{4}}{x^{4} -1}dx = x + \frac{1}{4} ln(x-1) - \frac{1}{4} ln(x+1)-\frac{1}{2} arctanx + c

You might be interested in
Help please. Just need to answer 29-31 then I’m done. Thanks
Free_Kalibri [48]

Answer:

29. See table below

30. See attached graph

31. The slope is m= 0.10

The slope represent the cost for every additional call minute.

Step-by-step explanation:

The cost is $0.5  first minute and $0.10 for any additional minutes

If c is the total cost of a call that last t minutes then;

c= 0.10t + 0.5-----where t is the time the call lasted

29. Use the equation above to create the table as;

t {x}            c{y}

1                  0.6

2                 0.7

3                 0.8

4                 0.9

5                 1.0

6                 1.1

The graph of this plot is as attached , where the coordinates are

{1,0.6} , {2,0.7} ,{3,0.8} ,{4,0.9} ,{5,1.0}, {6,1.1}

The slope can be found using the formula;

m=Δy/Δx

m= 1.1 - 0.6 / 6-1

m= 0.5 / 5 = 0.10

The slope represent the cost for every additional call minute.

5 0
3 years ago
Please help me with this
ZanzabumX [31]

Answer:

300•0.15•6

Step-by-step explanation:


8 0
3 years ago
Find the equation of a line that passes through the point (2,-5) and is perpendicular to the y-axis. Do not put any spaces in yo
Allushta [10]

Answer:

<em>y=-5</em>

Step-by-step explanation:

<u>Equation of the line</u>

Lines can be expressed in several forms. The slope-intercept form of the line is:

y = mx + b

Where m is the slope of the line and m is the y-intercept.

If the line is perpendicular to the y-axis, then it's a horizontal line. Horizontal lines have slope m=0, thus, the equation is:

y = b

To find the value of b, just substitute the given point:

-5=b

The equation of the line is:

y=-5

3 0
2 years ago
Which ordered pairs in the form (x, y)(x, y) are solutions to the equation
Scilla [17]
4x - 5y = 24

4(6) - 5(0) = 24
24 - 0 = 24
24 = 24 (correct)....so (6,0) is a solution

4(1) - 5(-4) = 24
4 + 20 = 24
24 = 24 (correct)...and (1,-4) is a solution
3 0
3 years ago
Read 2 more answers
A painter earns $15 per hour.What is the minimum number of hours he must work to earn at least $200?
umka21 [38]

Let's represent h with the number of hours:

15h ≥ 200

h ≥ 200/15

h ≥ 40/3

h ≥ 13 1/3

He has to work at least 13 1/3 hours.

4 0
3 years ago
Other questions:
  • 0.0985 rounded to the nearest tenth<br><br> Need fast answersss!!
    10·1 answer
  • Which of the following functions involves exponential decay?
    7·1 answer
  • The length of a rectangle is twice it's width. If the perimeter of the rectangle is 58yd find its length and widtg
    15·1 answer
  • -
    5·1 answer
  • If two lines are perpendicular, which statement must be true? A) The product of their slopes is 1. B) Their slopes are opposites
    7·1 answer
  • A group of eight friends (five girls and three boys) plans to watch a movie but they have only five tickets how many different c
    11·1 answer
  • 4a+(-8), if a= -2 PLZ ANSWER ASAP
    14·1 answer
  • On graph paper, draw diagrams of a+a+a+a and 4a when a is 1, 2, and 3. What do you notice?
    10·2 answers
  • El triple del área de un cuadrado menos seis veces la medida de su lado es igual a 0. ¿Cuánto mide por lado el cuadrado?
    9·1 answer
  • Mel has a bag containing white and yellow marbles.
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!