Example of partial fraction decomposition, Mathematics

Assignment Help:

Example of Partial Fraction Decomposition

Evaluate the following integral.

∫ (3x+11 / x2-x-6) (dx)

Solution:

The 1st step is to factor the denominator so far as possible and acquire the form of the partial fraction decomposition.  By doing this gives,

3x + 11 = (x - 3) (x + 2)

A / x - 3 + B/ x + 2

The other next step is to actually add the right side back up.

3x + 11 / (x - 3) (x + 2)

 A (x + 2) + B (x - 3) / (x - 3) (x + 2)

Here now, we require to choose A and B that is why the numerators of these two are equal for every x.  To do this we'll require to set the numerators equal.

3x + 11 = A (x + 2) + B (x - 3)

Note: In several problems we will go straight from the common form of the decomposition to this step and not bother with in fact adding the terms back up. The one point to adding the terms is to get the numerator and we can obtain that without actually writing down the results of the addition.

At this point we contain one of two ways to proceed.  One way will all time work, but is frequently more work.  The other, when it won't always work, is frequently quicker when it does work. In this case both will work and thus we'll use the quicker way for this instance.

What we are going to do at this time is to notice that the numerators have to be equal for any x that we would choose to use.  Especially the numerators must be equal for plug these in and see what we obtain.

x = -2                           5 = A (0) + B (-5)     ⇒        B = -1

x = 3                              20 = A (5) + B (0)     ⇒        A = 4

Thus, by carefully picking the x's we acquire the unknown constants to quickly drop out. Note: These are the values we claimed they would be above.

At this point there really is not a whole lot to do except the integral.

42_Example of Partial Fraction Decomposition.png

Remind that to do this integral we first split it up into two integrals and after that used the substitutions,

U = x-3                        

V = x+2

On the integrals to get the last answer


Related Discussions:- Example of partial fraction decomposition

Determines the first four derivatives of y = cos x, Example    determines t...

Example    determines the first four derivatives for following.                                                                  y = cos x Solution: Again, let's just do so

Reduced Row-Echelon Form, The augmented matrix from a system of linear equa...

The augmented matrix from a system of linear equations has the following  reduced row-echelon form (a)  How many equations are there in the system?  (b)  How many variab

Calculate plurality based on the number of voters and candid, Consider an e...

Consider an election with 721 voters. A) If there are 5 candidates, at least x votes are needed to have a plurality of the votes. Find x. B) Suppose that at least 73 votes are n

X and Y Intercepts, Find the x and y intercepts for the following equations...

Find the x and y intercepts for the following equations: 3y=3x -y=-x-4 2x+3y=6 y=5

Identify the surface for the equation , Identify the surface for each of th...

Identify the surface for each of the subsequent equations. (a) r = 5 (b) r 2 + z 2 = 100 (c) z = r Solution (a)  In two dimensions we are familiar with that this

Equilibrium solutions, In the earlier section we modeled a population depen...

In the earlier section we modeled a population depends on the assumption that the growth rate would be a constant. Though, in reality it doesn't make much sense. Obviously a popula

Minimizes the sum of the two distance, The value of y that minimizes the su...

The value of y that minimizes the sum of the two distances from (3,5) to (1,y) and from (1,y) to (4,9) can be written as a/b where a and b are coprime positive integers. Find a+b.

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd