The quotient of seven and a number added to ten

Please, I need help in this ??

nignag [31]

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,

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

4 0

3 years ago

Help me please and explain. <br><br>FIND THE AREA OF THE UNSHADED REGION

bagirrra123 [75]

To solve this first find the area of each of the white shapes inside the rectangle

The formula for the area of a circle is:

A = πr²

In this case we know that the radius is 2

Plug what you know into the area of a circle formula:

A = π2²

A = π4

A = 4π

A ≈ 12.57

The formula for the area of a square is:

A = Length x Height

In this case we know that both length and height are 3

Plug what you know into the formula for area of a square:

A = 3 x 3

A = 9

To find the area of the UNSHADED region simply add together the area of the circle (12.57) with the area of the square (9):

12.57 + 9 ≈ 21.57

To find the area of the SHADED region find the total area of the surrounding rectangle then subtract the area of unshaded area from the total area of the surrounding rectangle.

The formula for the area of a rectangle is:

A = Length x Height

The length is 10 and the height is 7

A = 10 x 7

A = 70

Shaded region:

70 - 21.57 = 48.43

Unshaded region: 21.57

Total area of surrounding rectangle: 70

Shaded region: 48.43

Hope this helped!

~Just a girl in love with Shawn Mendes

5 0

3 years ago

Read 2 more answers

Read carefully and choose the name of the student who made the correct statement ..... HELPPP hurry

Arlecino [84]

Answer:

Tyra.

note that the salary for only one car is less at dealership B, but significantly more when you get to 10 cars. that indicates that the base salary is less, but the commissions are higher.

7 0

2 years ago

which will have a higher effective interest rate a payday loan for $1700 that is due in 16 days with a fee of $100, or a payday

Fantom [35]

1700 that is due in 14 days with a fee of 100$ because it’s a shorter time period

7 0

2 years ago

Read 2 more answers

Which of the following graphs shows a pair of lines that represent the equations with a solution (−5, 2)? (PLEASE HELP ASAP, DUE

gizmo_the_mogwai [7]

Using a system of equations, it is found that the third graph shows a pair of lines that represent the equations with a solution (−5, 2).

<h3>What is a system of equations?</h3>

A system of equations is when two or more variables are related, and equations are built to find the values of each variable.

The solution of the system on a graph is the intersection of two lines. The third graph has an intersection at (-5,2), hence it is the answer to this question.

More can be learned about a system of equations at brainly.com/question/24342899

#SPJ1

5 0

1 year ago