- #1

Xcron

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The problem is:

[tex]\int\frac{4x^4-43x^3+200x^2-442x+383}{(x-3)(x^2-6x+13)}dx[/tex]

It comes with a small help thing that says: [tex](x-3)(x^2-6x+13) =

x^3-9x^2+31x-39[/tex]

Ok so the first thing I did was use long division and I got [tex]4x-7[/tex] as the answer of the long division. Next I proceeded to do:

[tex]4x^4-43x^3+200x^2-442x+383 = A(x^2-6x+13) + (Cx+D)(x-3)[/tex].

Next, I plugged in 3 for the value of x so that I could solve for A and I got its value to be 5. Next, I plugged in 0 for the value of x so that I could get A and D alone and solve for D. I got its value to be -106, which is horribly wrong for some reason, I think. Next I plugged in 1 for the value of x and solved for C nd its value came out to be 75, which is also horribly wrong probably..I made a mistake somewhere in there I think...after that, I set up the new integral:

[/tex]\int\(4x-7+\frac{5}{x-3}+\frac{75x-106}{x^2-6x+13})dx[/tex].

I solved that and got my final answer to be:

[tex]2x^2-7x+5\ln|x-3|+\frac{75}{2}\ln|x^2-6x+13|+

\frac{119}{2}\arctan(\frac{x-3}{2})+C[/tex].

This answer is wrong and the correct answer is:

[tex]2x^2-7x+5\ln|x-3|+4\ln|x^2-6x+13|+

\frac{9}{2}\arctan(\frac{x-3}{2})+C[/tex].

If I am making some kind of mistake, then I must be inherently making it because all my other answers to problems with quadratic factors is coming out wrong...please help me.