I am currently reading a book on mathematical methods which presents the following line integral problem without a solution:
$F = \frac{1}{1+x+y^2+z^2} \textbf{i} + \frac{2y}{1+x+y^2+z^2} \textbf{j} + \frac{2z}{1+x+y^2+z^2} \textbf{k}$
Find the line integral of F from (0,0,0) to (1,1,1) along C, where C is defined as the line from (0,0,0) to (1,1,0) followed by the line from (1,1,0) to (1,1,1).
I have tried parameterising by t in two parts, setting variables to 0 where appropriate. This gives me:
$\int F(r(t))=\int_{0}^{1} \frac{2t+1}{t^2+t+1} dt + \int_{0}^{1} \frac{2t}{t^2+1} dt \\ = [ln|t^2+t+1|]_0^1 + [ln|t^2+1|]_0^1 \\ = ln(3)+ln(2)$
Am I correct in doing this?
I'm assuming you're using $\vec r_1(t) = (t,t,0)$ with $0 \le t \le 1$ as the parametrisation for the first line and that integral looks alright.
For the parametrisation of the second line going from $(1,1,0)$ to $(1,1,1)$, you want something like $\vec r_2(t) = (1,1,t)$ with $0 \le t \le 1$ but then you would find something different for the second integral because: $$F(\vec r_2(t)) \cdot \vec r'_2(t) = \left(\cdots,\cdots,\frac{2t}{1+1+1^2+t^2}\right) \cdot (0,0,1) = \frac{2t}{3+t^2}$$ which leads to $$\int_{C_2} F \cdot ds = \int_0^1 \frac{2t}{3+t^2} \, dt = \left[ \ln\left|3+t^2\right| \right]_0^1= \ldots \ne \ln 2$$