Question

You will integrate the following rational function: ∫ −2 x2+19 x−70(x−5) (x2−6 x+10) dx To do this, start by writing −2 x2+19 x−70(x−5) (x2−6 x+10) = Ax−5+B x+Cx2−6 x+10 (a) Put the right-hand side over a common denominator. Enter the numerator of the result. Ax^2-6Ax+10A+Bx^2-5Bx+Cx-5C Preview (b) Expand the numerator. It is a quadratic in x , which must equal the numerator of the original function. By equating the coefficients you should be able to formulate linear equation in A , B and C . In each case give an equation with the constant part on the right, for example A+2B=3. What equation can you deduce from the coefficients of: x2 : A+B=-2 Preview x : -6A-5B+C=19 Preview constants: 10A-5C=-70 Preview (c) Solve these three equations, and enter the solution in the form A,B,C A,B,C= -5,3,4 Preview (d) The last term is in the form B x+Cx2−6 x+10 . Find constants K and J such that the numerator B x+C=K (2 x−6)+J . Enter in the form K,J K,J= 3/2,13 Preview (e) Separating out the K and J terms, you now have a term of the form Jx2−6 x+10 . By completing the square in the denominator you should be able to express this in the form J(x−r)2+1 for some constant r . r= 3 Preview (f) Your rational function is now in the form Ax−5+K (2 x−6)x2−6 x+10+J(x−r)2+1 . Evaluate the integral. You might need to use simple substitutions for the second and third terms. Use a constant of integration c. Don’t forget that the integral of 1x is ln(|x|) , and the syntax for this is ln(abs(x)). Integral = Preview

Name: you will integrate the following rational function 2 x219 x70x5 x26 x10 dx to do this start by writing 2 x219 x70x5 x26 x10 ax5b xcx26 x10 a put the right hand side over a common denominator 33934
Uploaded: 2024-04-22T17:40:02-08:00
Duration: 7 min 22 s
Channel: Amelia Hardy
Description: you will integrate the following rational function 2 x219 x70x5 x26 x10 dx to do this start by writing 2 x219 x70x5 x26 x10 ax5b xcx26 x10 a put the right hand side over a common denominator 33934

          You will integrate the following rational function:
∫
  −2 x2+19 x−70(x−5) (x2−6 x+10)
   dx
To do this, start by writing  −2 x2+19 x−70(x−5) (x2−6 x+10)
  =  Ax−5+B x+Cx2−6 x+10
(a) Put the right-hand side over a common denominator. Enter the numerator of the result.
A*x^2-6*A*x+10*A+B*x^2-5*B*x+C*x-5*C
Preview  
(b) Expand the numerator. It is a quadratic in  x
 , which must equal the numerator of the original function.
By equating the coefficients you should be able to formulate linear equation in  A
 ,  B
   and  C
 .
In each case give an equation with the constant part on the right, for example A+2*B=3.
What equation can you deduce from the coefficients of:
x2
  : 
A+B=-2
Preview  
x
 : 
-6*A-5*B+C=19
Preview  
constants: 
10*A-5*C=-70
Preview  
(c) Solve these three equations, and enter the solution in the form  A,B,C
A,B,C=
  
-5,3,4
Preview  
(d) The last term is in the form  B x+Cx2−6 x+10
 .
Find constants  K
  and  J
   such that the numerator  B x+C=K (2 x−6)+J
 .
Enter in the form  K,J
K,J=
  
3/2,13
Preview  
(e) Separating out the  K
 and  J
 terms, you now have a term of the form  Jx2−6 x+10
 .
By completing the square in the denominator you should be able to express this in the form  J(x−r)2+1
  for some constant  r
 .
r=
  
3
Preview  
(f) Your rational function is now in the form  Ax−5+K (2 x−6)x2−6 x+10+J(x−r)2+1
 .
Evaluate the integral. You might need to use simple substitutions for the second and third terms.
Use a constant of integration c.
Don’t forget that the integral of  1x
   is  ln(|x|)
 , and the syntax for this is ln(abs(x)).
Integral = 
Preview

Added by Gregorio H.

Question

Please give Ace some feedback