You will integrate the following rational function

Question

You will integrate the following rational function:

∫ (−2 x^2 −24 x +138)/(x−5)^2 (x+3) dx

To do this, start by writing (−2 x^2 −24 x +138)/(x−5)^2 (x+3) = A/x−5 + B/(x−5)^2 + C/x+3

(a) Put the right-hand side over a common denominator. Enter the numerator of the result.

(b) Expand the numerator. By equating the coefficients you should be able to formulate linear equation in A, Band 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 :

x^2 = ?

x = ?

constant = ?

c) Solve these three equations, and enter the solution in the form A,B,C

(d) Use these values, and the partial fraction form, to find the integral. Use a constant of integration F.

Don’t forget that the integral of 1/x is ln(abs(x)), not just ln(x).

Do not combine separate terms using the log laws, this can result in functions with different domains so Maple may not view them as equal.

Answer 1

A/(x-5)+B/(x-5)²+C/(x+3) = ((x-5)(x+3)A+(x+3)B-(x-5)²+(x-5)²C)/((x-5)²(x-3))

x²: A+C=-2

x: -2A+B-10C = -24

const: -15A+3B+25C = 138

A = -5

B= -4

C = 3

∫-5/(x+5) dx + ∫-4/(x+5)² dx + ∫3/(x+3) dx = -5ln|x-5|+4/(x-5)+3ln|x+3| + F

Answer 2

desmos.com/calculator/ksabtpzpip

The above graph shows the original function and the partial fraction decomposition of that function (identical).