To integrate ∫csc^2x dx, we use u-substitution. Let u = cot(x) and du = -csc^2x dx. Then:
∫csc^2x dx = ∫-du/u^2
We can integrate this by using the power rule of integration:
∫-du/u^2 = 1/u + C
Substituting back in for u, we get:
∫csc^2x dx = 1/cot(x) + C
Simplifying by using the reciprocal identity, we get:
∫csc^2x dx = tan(x) + C
Thus, the antiderivative of csc^2x is tan(x) + C.
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