Saturday, 4 March 2017

calculus - How to integrate intfrac3x+2x2x2dx




This is the indefinite integral I have to evaluate: x3x2x2dx
so by using the long division on polynomials technique, I got to: x22+x+3x+2x2x2dx
How do I continue from here? I thought of using "integration of rational functions" but it didn't work.


Answer



Specifically, you have that 3x+2x2x2=Ax2+Bx+1



The usual technique reveals that A=83 and B=13







Hence x3x2x2=x+1+3x+2(x2)(x+1)=x+1+83(x2)+13(x+1)



So that x3x2x2dx=x+1+3x+2x2x2=x+1+83(x2)+13(x+1)dx



Hence x3x2x2=x+x22+83ln|x2|+13ln|x+1|+C


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