## Integration Using Partial Fractions

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

\[\frac{3x-2}{(x^2+1)(x-2)^3}= \frac{Ax+B}{x^2+1}+ \frac{C}{x-2} + \frac{D}{(x-2)^2} + \frac{E}{(x-2)^3}\]

.