csc^2x dx
-cotx + c
To integrate csc^2(x) dx, we can use substitution. Let u = cos(x) which means du/dx = -sin(x).
Substituting u = cos(x) and du/dx = -sin(x) into the integral, we get:
∫csc^2(x) dx = ∫(1/sin^2(x)) dx
= ∫(1/(1-cos^2(x))) (-sin(x) dx) (using the substitution u=cos(x), sin^2(x) = 1 – cos^2(x))
= -∫(1/(1-u^2)) du
= -1/2 ln |(1+u)/(1-u)| + C
= -1/2 ln |(1+cos(x))/(1-cos(x))| + C
Therefore, the final answer is -1/2 ln |(1+cos(x))/(1-cos(x))| + C.
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