Evaluating Partial Fractions

Quote:

Originally Posted by shogunhd

Evaluate the Integral

(x)/(x^2+7x+6) from 0 to 4

Since I don't use partial fractions technique frequently, you can do this as follows:

$\int_0^4 {\frac{x} {{x^2 + 7x + 6}}\,dx} = \frac{1} {5}\int_0^4 {\frac{{6(x + 1) - (x + 6)}} {{(x + 1)(x + 6)}}\,dx} .$

So $\int_0^4 {\frac{x} {{x^2 + 7x + 6}}\,dx} = \frac{1} {5}\left( {6 \cdot \ln \left| {x + 6} \right|\Big|_0^4 - \ln \left| {x + 1} \right|\Big|_0^4} \right).$

Hence $\int_0^4 {\frac{x} {{x^2 + 7x + 6}}\,dx} = \frac{1} {5}\left( {6\ln 10 - 6\ln 6 - \ln 5} \right).$

Now here's the make-up:

$\begin{aligned} \frac{1} {5}\left( {6\ln 10 - 6\ln 6 - \ln 5} \right) &= \frac{1} {5}\left( {6\ln 2 + 6\ln 5 - 6\ln 2 - 6\ln 3 - \ln 5} \right)\\ &= \frac{1} {5}\left( {5\ln 5 - 6\ln 3} \right)\\ &= \ln 5 - \frac{6} {5}\ln 3,\end{aligned}$

which gives the desired answer.