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

Solution to a differential equation, A solution to a differential equation ...

A solution to a differential equation at an interval α Illustration 1:   Show that y(x) = x -3/2 is a solution to 4x 2 y′′ + 12xy′ + 3 y = 0 for x > 0. Solution : We'll

Example of 3-d coordinate system, Example of 3-D Coordinate System Exam...

Example of 3-D Coordinate System Example: Graph x = 3 in R, R 2 and R 3 .   Solution In R we consist of a single coordinate system and thus x=3 is a point in a 1-D co

Explain how we converting fractions to percents, Explain how we Converting ...

Explain how we Converting Fractions to Percents ? To convert a fraction to a percent: 1. Convert the fraction to a decimal using long division. 2. Move the decimal point two p

Partial fractions - integration techniques, Partial Fractions - Integration...

Partial Fractions - Integration techniques In this part we are going to take a look at integrals of rational expressions of polynomials and again let's start this section out w

Operations Research inventory , A firm buys a product using the price sched...

A firm buys a product using the price schedule given in the table: The company estimate holding costs at 10% of the purchase price per year and ordering costs at $40 per order .

Linear function, find the temperature at which the celsius and farhenheit t...

find the temperature at which the celsius and farhenheit temperatures are numerically equl

Determine principal strains and direction , A 100 by 150 mm rectangular pla...

A 100 by 150 mm rectangular plate is deformed as shown in the following figure. All dimensions shown in the figure are in millimeters.  Determine at point Q: (a) the strain compone

Finding the equation of a line, Finding the Equation of a line, Given a Slo...

Finding the Equation of a line, Given a Slope and a Point ? Find the equation of a line with slope m = 2, which passes through the point (-1, -3). Solution: Use the po

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