∫csc(x)dx
-lnIcsc(x)+cot(x)I+c
We can solve this integral using u-substitution.
Let u = sin(x), then du/dx = cos(x) and dx = du/cos(x).
Substituting this into the integral:
∫csc(x)dx = ∫csc(x)(cos(x)/cos(x))dx
= ∫(1/u)(du/dx)dx
= ∫du/u
= ln|u| + C
Now, substituting back in for u:
ln|sin(x)| + C
So, the final answer is ln|sin(x)| + C, where C is the constant of integration.
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