Thread: Finding the intergral of the function?

1. Finding the intergral of the function?

Ok to start off with, I have the integral (5x^2+12x+6)/((x+2)(x+1)^2) between the upper limit of 1 and lower limit of 0. I need to solve it using partial fractions. I have worked out values, but am not sure if they are true. I found A=2, B3 and C=-1, which creates the fractions 2/(x+2)+3/(x+1)-1/(x+1)^2. How do I integrate these and what is the answer when the limits are placed?

2. Yes, that's the partial fraction descomposition, good job. Now, we have to find $\int_{0}^{1}{\left( \frac{2}{x+2}+\frac{3}{x+1}-\frac{1}{(x+1)^{2}} \right)\,dx}.$ The integration of these are pretty easy; for example, consider the first integral and put $u=x+2.$ The important thing here is you to recognize that $\int{\frac{du}{u}}=\ln \left| u \right|+k.$ (For the first two, the third one requires substitution and the power rule application.) Once done this, the integral equals $\left. 2\ln \left| x+2 \right|+3\ln \left| x+1 \right|+\frac{1}{x+1} \right|_{0}^{1}=2\ln 3+3\ln 2+\frac{1}{2}-(2\ln 2+1).$