Integration Using Partial Fractions

We can integrate fractions where the denominator is a polynomial using partial fractions, which involves separating the fraction into simpler terms and integrating each one.
Example:  
\[\int \frac{2}{x^2+3x+2} dx\]
.
We solve for  
\[A\]
  and  
\[B\]
  the expression
\[\frac{2}{x^2+3x+2}= \frac{A}{x+1} + \frac{B}{x+2} \rightarrow 2=A(x+2)+B(x+1)\]

Let  
\[x=-2\]
  then
\[2=-B \rightarrow B=-1\]

Let  
\[x=-2\]
  then
\[2=A\]

The integral becomes  
\[\int \frac{2}{x+1} - \frac{2}{x+2} dx\]
  and the result is  
\[2 ln(x_1)-2ln(x+2)= 2 ln( \frac{x+1}{x+2} )\]
.
In general for each factor in the denominator we must assume a numerator of degree one less than the degree of the numerator, and a repeated factor in the denominator gives rise to terms, with the denominator in each term being powers of the repeated factor.
Example  
\[\frac{3x-2}{(x^2+1)(x-2)^3}= \frac{Ax+B}{x^2+1}+ \frac{C}{x-2} + \frac{D}{(x-2)^2} + \frac{E}{(x-2)^3}\]
.

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