Integrating Rational Functions with Complex Quadratics and Partial Fractions

[Master J]

I want to integrate:

1/[(x + 1)*(x^2 + x +1)] dx

Now the quadratic has complex routes, and we have not done any integration with that yet, so I broke it up into its partial fractions.

A/(x +1) + (Bx + C)/(x^2 + x +1)

But I cannot seem to find the numbers A B C. mamybe I am just missing something real obvious?? Any pointers in the right direction? Cheers guys.

PS. Is the proof of the theory that you can break up fractions like that beyond a first year math for science course?

 

[HallsofIvy]

I have no idea what a "math for science" course is, but the proof that there exist numbers A, B, C that will work doesn't involve anything more than basic algebra.

Here you want to find A, B, C so that
$$\frac{1}{(x+1)(x^2+ x+1)}= \frac{A}{x+1}+ \frac{Bx+ C}{x^2+ x+ 1}$$
Multiply both sides by (x+1)(x²+ x+ 1) to get
$$1= A(x^2+ x+ 1)+ (Bx+ C)(x+1)$$
If you let x= -1, that reduces to 1= A.

I expect you had already done that. The problem is that that there is no value of x that makes x²+ x+ 1= 0. You cannot "reduce" the equation that easily but since the equation is true for all x, you can still get two equations for B and C by letting x be any number you want. I would suggest putting x= 0 and x= 1 into the equation.
If x= 0, the equation becomes 1= A+ C and you already know A. If x= 1, the equation becomes 1= 3A+ (B+ C)(2) and you already know A and C.

Another method that always works is to multiply out the right side and combine "like powers"
$$1= A(x^2+ x+ 1)+ (Bx+ C)(x+1)$$
$$1= Ax^2+ Ax+ A+ Bx^2+ Bx+ Cx+ C$$
$$1= (A+ B)x^2+ (A+ B+ C)x+ (A+ C)$$
Since that must be true for all x, corresponding coefficients must be the same: A+ B= 0, A+ B+ C= 0, A+ C= 1.

To integrate the term with x²+ x+ 1 in the denominator, complete the square to get (x+ 1/2)²+ 3/4 and let u= x+ 1/2.

 

[Master J]

Thank you.

I had already got A=1, and C=0. I don't know how I didn't get B. I guess I was a bit tired and lost track :shy:

I was just wondering about the other part. My Mathematics course is part of a science course, so sometimes proofs aren't done, like they would be in a pure math course.

 

[Master J]

On this question again, how do I integrate the second part if there is also an x in the top?

 

[HallsofIvy]

On this question again, how do I integrate the second part if there is also an x in the top?

As I said before, complete the square in the denominator so it is (x+ 1/2)²+ 3/4, then let u= x+ 1/2. You will have something of the form $(u+ c)/(u^2+ 3/4)$. Separate that as $u/(u^2+ 3/4)+ c/(u^2+ 3/4)$. The first is easy: let v= u²+ 3/4. For the second remember that the derivative of arctan(x) is 1/(x²+ 1).