The problem I have is:
Calculate the integral (here $a < -1$)
$$\int_a^{-1} \left( t+\frac{1}{t}\right)\, dt$$
What I tried:
- Looked up examples, but couldn't find any where one of the limits of integration was given as an interval like it is in this problem. I might not be using the correct terminology, so am stuck at the start.
First we find the anti-derivative: $$ \int\left(t + \frac{1}{t}\right)dt = \int t dt + \int \frac{dt}{t} = \frac{t^2}{2} + \ln |t| + c. $$ Thus, for $c<d\le -1$, $$ \int_c^d \left(t + \frac{1}{t}\right)dt = \left[ \frac{t^2}{2} + \ln |t| \right]_c^d = \left[ \frac{d^2}{2} + \ln |d| - \frac{c^2}{2} - \ln |c|\right] = \frac{d^2-c^2}{2} + \ln \left|\frac{d}{c}\right|. $$
Can you apply this to your specific problem?