So I consider the integral:
$$\int_{\pi/6}^\frac{\pi}{2}\csc\left(t\right)\cot\left(t\right)\space \text{dt}$$
In my notes I have: $$=-\csc\left(t\right)+C$$
Then I take the limit of $\csc(t)$:
$$\lim_{t\rightarrow\frac{\pi}{6}}-\csc(t)=-2$$
$$\lim_{t\rightarrow\frac{\pi}{2}}-\csc(t)=-1$$
Then I subtract $F(b)-F(a)$,
$-1-(-2)=1$ but I don't know how I arrived at $-\csc(t)+C$
Best Answer
You cannot split up your integral like you have tried in your original post. In order to evaluate this, notice that $$\frac{d}{dx} (-\csc x) = \csc x \cot x$$ Therefore, $\int\csc x \cot x = -\csc x + C$. Then you use the Fundamental Theorem of Calculus, as you have noted.