How to integrate csc^2(x)?

To integrate csc^2(x), use the substitution u = cot(x) and the formula ∫du/(1+u^2) = arctan(u) + C.

To integrate csc^2(x), we can use the substitution u = cot(x). Then, we have du/dx = -csc^2(x), which means that csc^2(x)dx = -du/u^2. Substituting this into the integral, we get:

∫csc^2(x)dx = -∫du/u^2

Using the formula ∫du/(1+u^2) = arctan(u) + C, we can integrate -∫du/u^2 as follows:

-∫du/u^2 = -(-1/u + C) = 1/u + C = 1/cot(x) + C = tan(x) + C

Therefore, the integral of csc^2(x) is tan(x) + C.