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
This are 3 different questions please help me
Marysya12 [62]
1. diginity
2.require
3.tuition
5 0
2 years ago
Which phrase best describes the translation from the graph y = (x – 5)2 + 7 to the graph of y = (x + 1)2 – 2?
Andreas93 [3]

The parent. function shifted up by 6 units to produce x + 1 and to the left by 9 units

<h3>What is translation?</h3>

This is a way of changing the position of an object on an xy-plane.

Given the parent function expressed as  y = (x – 5)^2 + 7

From the resulting image after translation, we can see that the parent. function shifted up by 6 units to produce x + 1 and to the left by 9 units

Learn more on translation here; brainly.com/question/12861087

#SPJ1

8 0
2 years ago
Need help...!! plzzz
kipiarov [429]
This is true sometimes and false sometimes, depending on the numbers.
3 0
3 years ago
Given the points (8,9) and (2,7) find the slope I need help
Debora [2.8K]

Answer:

1/3

Step-by-step explanation:

7-9=-2

2-8=-6

2/6 is 1/3 when simplified

6 0
3 years ago
b. Two events are dependent if the occurrence of one event changes to occurrence of the second event. True or False
Naddika [18.5K]

Answer:

true

Step-by-step explanation:

6 0
3 years ago
Read 2 more answers
Other questions:
  • A fair dice is rolled. Work out the probability of getting a factor of 12.
    15·2 answers
  • . Hot dogs come in packs of 8. Hot dog rolls come in packs of 12. What is the least number of packs of each Shawn should buy to
    9·2 answers
  • A rectangular prism with length \ellℓell, width www, and height hhh has a volume of \ell w hℓwhell, w, h. What is the volume of
    14·2 answers
  • suppose you have 192 marbles in groups of 15 marbles each.Find the number of groups of marbles that you have .Write the quotient
    10·2 answers
  • Please help me !!! A question for Pre- Calculus Conic Sections!
    11·1 answer
  • A room covers 299.6 square feet. A painter paints the room in three days. The first day he covers 2/7 of the area. The next day
    9·2 answers
  • X + 4 + 3x = 72 how did you get this answer?
    8·2 answers
  • If tests are worth 60% of your grade, and you have a 76.8% in that class, and you fail that test, what will be your grade?
    12·1 answer
  • What is the equation of the line in slope-intercept form?
    7·1 answer
  • $7.65 for 1.8 ibs of gummy bears
    15·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!